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Theorem List for Metamath Proof Explorer - 1601-1700   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremexintr 1601 Introduce a conjunct in the scope of an existential quantifier. (Contributed by NM, 11-Aug-1993.)
 |-  ( A. x (
 ph  ->  ps )  ->  ( E. x ph  ->  E. x ( ph  /\  ps )
 ) )
 
Theoremalsyl 1602 Theorem *10.3 in [WhiteheadRussell] p. 150. (Contributed by Andrew Salmon, 8-Jun-2011.)
 |-  ( ( A. x ( ph  ->  ps )  /\  A. x ( ps 
 ->  ch ) )  ->  A. x ( ph  ->  ch ) )
 
1.4.4  Axiom scheme ax-17 (Distinctness) - first use of $d
 
Axiomax-17 1603* Axiom of Distinctness. This axiom quantifies a variable over a formula in which it does not occur. Axiom C5 in [Megill] p. 444 (p. 11 of the preprint). Also appears as Axiom B6 (p. 75) of system S2 of [Tarski] p. 77 and Axiom C5-1 of [Monk2] p. 113.

(See comments in ax17o 2096 about the logical redundancy of ax-17 1603 in the presence of our obsolete axioms.)

This axiom essentially says that if  x does not occur in  ph, i.e.  ph does not depend on  x in any way, then we can add the quantifier  A. x to  ph with no further assumptions. By sp 1716, we can also remove the quantifier (unconditionally). (Contributed by NM, 5-Aug-1993.)

 |-  ( ph  ->  A. x ph )
 
Theorema17d 1604* ax-17 1603 with antecedent. Useful in proofs of deduction versions of bound-variable hypothesis builders. (Contributed by NM, 1-Mar-2013.)
 |-  ( ph  ->  ( ps  ->  A. x ps )
 )
 
Theoremnfv 1605* If  x is not present in  ph, then  x is not free in  ph. (Contributed by Mario Carneiro, 11-Aug-2016.)
 |- 
 F/ x ph
 
Theoremnfvd 1606* nfv 1605 with antecedent. Useful in proofs of deduction versions of bound-variable hypothesis builders such as nfimd 1761. (Contributed by Mario Carneiro, 6-Oct-2016.)
 |-  ( ph  ->  F/ x ps )
 
Theoremalimdv 1607* Deduction from Theorem 19.20 of [Margaris] p. 90. (Contributed by NM, 3-Apr-1994.)
 |-  ( ph  ->  ( ps  ->  ch ) )   =>    |-  ( ph  ->  (
 A. x ps  ->  A. x ch ) )
 
Theoremeximdv 1608* Deduction from Theorem 19.22 of [Margaris] p. 90. (Contributed by NM, 27-Apr-1994.)
 |-  ( ph  ->  ( ps  ->  ch ) )   =>    |-  ( ph  ->  ( E. x ps  ->  E. x ch ) )
 
Theorem2alimdv 1609* Deduction from Theorem 19.22 of [Margaris] p. 90. (Contributed by NM, 27-Apr-2004.)
 |-  ( ph  ->  ( ps  ->  ch ) )   =>    |-  ( ph  ->  (
 A. x A. y ps  ->  A. x A. y ch ) )
 
Theorem2eximdv 1610* Deduction from Theorem 19.22 of [Margaris] p. 90. (Contributed by NM, 3-Aug-1995.)
 |-  ( ph  ->  ( ps  ->  ch ) )   =>    |-  ( ph  ->  ( E. x E. y ps  ->  E. x E. y ch ) )
 
Theoremalbidv 1611* Formula-building rule for universal quantifier (deduction rule). (Contributed by NM, 5-Aug-1993.)
 |-  ( ph  ->  ( ps 
 <->  ch ) )   =>    |-  ( ph  ->  (
 A. x ps  <->  A. x ch )
 )
 
Theoremexbidv 1612* Formula-building rule for existential quantifier (deduction rule). (Contributed by NM, 5-Aug-1993.)
 |-  ( ph  ->  ( ps 
 <->  ch ) )   =>    |-  ( ph  ->  ( E. x ps  <->  E. x ch )
 )
 
Theorem2albidv 1613* Formula-building rule for 2 universal quantifiers (deduction rule). (Contributed by NM, 4-Mar-1997.)
 |-  ( ph  ->  ( ps 
 <->  ch ) )   =>    |-  ( ph  ->  (
 A. x A. y ps 
 <-> 
 A. x A. y ch ) )
 
Theorem2exbidv 1614* Formula-building rule for 2 existential quantifiers (deduction rule). (Contributed by NM, 1-May-1995.)
 |-  ( ph  ->  ( ps 
 <->  ch ) )   =>    |-  ( ph  ->  ( E. x E. y ps 
 <-> 
 E. x E. y ch ) )
 
Theorem3exbidv 1615* Formula-building rule for 3 existential quantifiers (deduction rule). (Contributed by NM, 1-May-1995.)
 |-  ( ph  ->  ( ps 
 <->  ch ) )   =>    |-  ( ph  ->  ( E. x E. y E. z ps  <->  E. x E. y E. z ch ) )
 
Theorem4exbidv 1616* Formula-building rule for 4 existential quantifiers (deduction rule). (Contributed by NM, 3-Aug-1995.)
 |-  ( ph  ->  ( ps 
 <->  ch ) )   =>    |-  ( ph  ->  ( E. x E. y E. z E. w ps  <->  E. x E. y E. z E. w ch ) )
 
Theoremalrimiv 1617* Inference from Theorem 19.21 of [Margaris] p. 90. (Contributed by NM, 5-Aug-1993.)
 |-  ( ph  ->  ps )   =>    |-  ( ph  ->  A. x ps )
 
Theoremalrimivv 1618* Inference from Theorem 19.21 of [Margaris] p. 90. (Contributed by NM, 31-Jul-1995.)
 |-  ( ph  ->  ps )   =>    |-  ( ph  ->  A. x A. y ps )
 
Theoremalrimdv 1619* Deduction from Theorem 19.21 of [Margaris] p. 90. (Contributed by NM, 10-Feb-1997.)
 |-  ( ph  ->  ( ps  ->  ch ) )   =>    |-  ( ph  ->  ( ps  ->  A. x ch ) )
 
Theoremnfdv 1620* Apply the definition of not-free in a context. (Contributed by Mario Carneiro, 11-Aug-2016.)
 |-  ( ph  ->  ( ps  ->  A. x ps )
 )   =>    |-  ( ph  ->  F/ x ps )
 
Theorem2ax17 1621* Quantification of two variables over a formula in which they do not occur. (Contributed by Alan Sare, 12-Apr-2011.)
 |-  ( ph  ->  A. x A. y ph )
 
1.4.5  Equality predicate; define substitution
 
Syntaxcv 1622 This syntax construction states that a variable  x, which has been declared to be a set variable by $f statement vx, is also a class expression. This can be justified informally as follows. We know that the class builder  { y  |  y  e.  x } is a class by cab 2269. Since (when  y is distinct from  x) we have  x  =  { y  |  y  e.  x } by cvjust 2278, we can argue that that the syntax " class  x " can be viewed as an abbreviation for " class  { y  |  y  e.  x }". See the discussion under the definition of class in [Jech] p. 4 showing that "Every set can be considered to be a class."

While it is tempting and perhaps occasionally useful to view cv 1622 as a "type conversion" from a set variable to a class variable, keep in mind that cv 1622 is intrinsically no different from any other class-building syntax such as cab 2269, cun 3150, or c0 3455.

For a general discussion of the theory of classes and the role of cv 1622, see http://us.metamath.org/mpeuni/mmset.html#class.

(The description above applies to set theory, not predicate calculus. The purpose of introducing 
class  x here, and not in set theory where it belongs, is to allow us to express i.e. "prove" the weq 1624 of predicate calculus from the wceq 1623 of set theory, so that we don't "overload" the  = connective with two syntax definitions. This is done to prevent ambiguity that would complicate some Metamath parsers.)

 class  x
 
Syntaxwceq 1623 Extend wff definition to include class equality.

For a general discussion of the theory of classes, see http://us.metamath.org/mpeuni/mmset.html#class.

(The purpose of introducing 
wff  A  =  B here, and not in set theory where it belongs, is to allow us to express i.e. "prove" the weq 1624 of predicate calculus in terms of the wceq 1623 of set theory, so that we don't "overload" the  = connective with two syntax definitions. This is done to prevent ambiguity that would complicate some Metamath parsers. For example, some parsers - although not the Metamath program - stumble on the fact that the  = in  x  =  y could be the  = of either weq 1624 or wceq 1623, although mathematically it makes no difference. The class variables  A and  B are introduced temporarily for the purpose of this definition but otherwise not used in predicate calculus. See df-cleq 2276 for more information on the set theory usage of wceq 1623.)

 wff  A  =  B
 
Theoremweq 1624 Extend wff definition to include atomic formulas using the equality predicate.

(Instead of introducing weq 1624 as an axiomatic statement, as was done in an older version of this database, we introduce it by "proving" a special case of set theory's more general wceq 1623. This lets us avoid overloading the  = connective, thus preventing ambiguity that would complicate certain Metamath parsers. However, logically weq 1624 is considered to be a primitive syntax, even though here it is artificially "derived" from wceq 1623. Note: To see the proof steps of this syntax proof, type "show proof weq /all" in the Metamath program.) (Contributed by NM, 24-Jan-2006.)

 wff  x  =  y
 
Theoremequs3 1625 Lemma used in proofs of substitution properties. (Contributed by NM, 5-Aug-1993.)
 |-  ( E. x ( x  =  y  /\  ph )  <->  -.  A. x ( x  =  y  ->  -.  ph ) )
 
Theoremspeimfw 1626 Specialization, with additional weakening to allow bundling of  x and  y. Uses only Tarski's FOL axiom schemes. (Contributed by NM, 23-Apr-2017.) (Proof shortened by Wolf Lammen, 5-Aug-2017.)
 |-  ( x  =  y 
 ->  ( ph  ->  ps )
 )   =>    |-  ( -.  A. x  -.  x  =  y  ->  ( A. x ph  ->  E. x ps )
 )
 
Theoremspimfw 1627 Specialization, with additional weakening to allow bundling of  x and  y. Uses only Tarski's FOL axiom schemes. (Contributed by NM, 23-Apr-1017.) (Proof shortened by Wolf Lammen, 7-Aug-2017.)
 |-  ( -.  ps  ->  A. x  -.  ps )   &    |-  ( x  =  y  ->  (
 ph  ->  ps ) )   =>    |-  ( -.  A. x  -.  x  =  y 
 ->  ( A. x ph  ->  ps ) )
 
Theoremax11i 1628 Inference that has ax-11 1715 (without  A. y) as its conclusion. Uses only Tarski's FOL axiom schemes. The hypotheses may be eliminable without one or more of these axioms in special cases. Proof similar to Lemma 16 of [Tarski] p. 70. (Contributed by NM, 20-May-2008.)
 |-  ( x  =  y 
 ->  ( ph  <->  ps ) )   &    |-  ( ps  ->  A. x ps )   =>    |-  ( x  =  y  ->  (
 ph  ->  A. x ( x  =  y  ->  ph )
 ) )
 
Syntaxwsb 1629 Extend wff definition to include proper substitution (read "the wff that results when  y is properly substituted for  x in wff  ph"). (Contributed by NM, 24-Jan-2006.)
 wff  [ y  /  x ] ph
 
Definitiondf-sb 1630 Define proper substitution. Remark 9.1 in [Megill] p. 447 (p. 15 of the preprint). For our notation, we use  [ y  /  x ] ph to mean "the wff that results from the proper substitution of  y for  x in the wff  ph." We can also use  [ y  /  x ] ph in place of the "free for" side condition used in traditional predicate calculus; see, for example, stdpc4 1964.

Our notation was introduced in Haskell B. Curry's Foundations of Mathematical Logic (1977), p. 316 and is frequently used in textbooks of lambda calculus and combinatory logic. This notation improves the common but ambiguous notation, " ph ( y ) is the wff that results when  y is properly substituted for  x in  ph ( x )." For example, if the original  ph ( x ) is  x  =  y, then  ph ( y ) is  y  =  y, from which we obtain that  ph ( x ) is  x  =  x. So what exactly does  ph ( x ) mean? Curry's notation solves this problem.

In most books, proper substitution has a somewhat complicated recursive definition with multiple cases based on the occurrences of free and bound variables in the wff. Instead, we use a single formula that is exactly equivalent and gives us a direct definition. We later prove that our definition has the properties we expect of proper substitution (see theorems sbequ 2000, sbcom2 2053 and sbid2v 2062).

Note that our definition is valid even when  x and  y are replaced with the same variable, as sbid 1863 shows. We achieve this by having  x free in the first conjunct and bound in the second. We can also achieve this by using a dummy variable, as the alternate definition dfsb7 2058 shows (which some logicians may prefer because it doesn't mix free and bound variables). Another version that mixes free and bound variables is dfsb3 1996. When  x and  y are distinct, we can express proper substitution with the simpler expressions of sb5 2039 and sb6 2038.

There are no restrictions on any of the variables, including what variables may occur in wff 
ph. (Contributed by NM, 5-Aug-1993.)

 |-  ( [ y  /  x ] ph  <->  ( ( x  =  y  ->  ph )  /\  E. x ( x  =  y  /\  ph )
 ) )
 
Theoremsbequ2 1631 An equality theorem for substitution. (Contributed by NM, 5-Aug-1993.)
 |-  ( x  =  y 
 ->  ( [ y  /  x ] ph  ->  ph )
 )
 
Theoremsb1 1632 One direction of a simplified definition of substitution. (Contributed by NM, 5-Aug-1993.)
 |-  ( [ y  /  x ] ph  ->  E. x ( x  =  y  /\  ph ) )
 
Theoremsbimi 1633 Infer substitution into antecedent and consequent of an implication. (Contributed by NM, 25-Jun-1998.)
 |-  ( ph  ->  ps )   =>    |-  ( [ y  /  x ] ph  ->  [ y  /  x ] ps )
 
Theoremsbbii 1634 Infer substitution into both sides of a logical equivalence. (Contributed by NM, 5-Aug-1993.)
 |-  ( ph  <->  ps )   =>    |-  ( [ y  /  x ] ph  <->  [ y  /  x ] ps )
 
1.4.6  Axiom scheme ax-9 (Existence)
 
Axiomax-9 1635 Axiom of Existence. One of the equality and substitution axioms of predicate calculus with equality. This axiom tells us is that at least one thing exists. In this form (not requiring that  x and  y be distinct) it was used in an axiom system of Tarski (see Axiom B7' in footnote 1 of [KalishMontague] p. 81.) It is equivalent to axiom scheme C10' in [Megill] p. 448 (p. 16 of the preprint); the equivalence is established by ax9o 1890 and ax9from9o 2087. A more convenient form of this axiom is a9e 1891, which has additional remarks.

Raph Levien proved the independence of this axiom from the other logical axioms on 12-Apr-2005. See item 16 at http://us.metamath.org/award2003.html.

ax-9 1635 can be proved from the weaker version ax9v 1636 requiring that the variables be distinct; see theorem ax9 1889.

ax-9 1635 can also be proved from the Axiom of Separation (in the form that we use that axiom, where free variables are not universally quantified). See theorem ax9vsep 4145.

Except by ax9v 1636, this axiom should not be referenced directly. Instead, use theorem ax9 1889. (Contributed by NM, 5-Aug-1993.) (New usage is discouraged.)

 |- 
 -.  A. x  -.  x  =  y
 
Theoremax9v 1636* Axiom B7 of [Tarski] p. 75, which requires that  x and  y be distinct. This trivial proof is intended merely to weaken axiom ax-9 1635 by adding a distinct variable restriction. From here on, ax-9 1635 should not be referenced directly by any other proof, so that theorem ax9 1889 will show that we can recover ax-9 1635 from this weaker version if it were an axiom (as it is in the case of Tarski).

Note: Introducing  x y as a distinct variable group "out of the blue" with no apparent justification has puzzled some people, but it is perfectly sound. All we are doing is adding an additional redundant requirement, no different from adding a redundant logical hypothesis, that results in a weakening of the theorem. This means that any future theorem that references ax9v 1636 must have a $d specified for the two variables that get substituted for  x and  y. The $d does not propagate "backwards" i.e. it does not impose a requirement on ax-9 1635.

When possible, use of this theorem rather than ax9 1889 is preferred since its derivation from axioms is much shorter. (Contributed by NM, 7-Aug-2015.)

 |- 
 -.  A. x  -.  x  =  y
 
Theorema9ev 1637* At least one individual exists. Weaker version of a9e 1891. When possible, use of this theorem rather than a9e 1891 is preferred since its derivation from axioms is much shorter. (Contributed by NM, 3-Aug-2017.)
 |- 
 E. x  x  =  y
 
Theoremspimw 1638* Specialization. Lemma 8 of [KalishMontague] p. 87. Uses only Tarski's FOL axiom schemes. (Contributed by NM, 19-Apr-2017.) (Proof shortened by Wolf Lammen, 7-Aug-2017.)
 |-  ( -.  ps  ->  A. x  -.  ps )   &    |-  ( x  =  y  ->  (
 ph  ->  ps ) )   =>    |-  ( A. x ph 
 ->  ps )
 
Theoremspimvw 1639* Specialization. Lemma 8 of [KalishMontague] p. 87. Uses only Tarski's FOL axiom schemes. (Contributed by NM, 9-Apr-2017.)
 |-  ( x  =  y 
 ->  ( ph  ->  ps )
 )   =>    |-  ( A. x ph  ->  ps )
 
Theoremspnfw 1640 Weak version of sp 1716. Uses only Tarski's FOL axiom schemes. (Contributed by NM, 1-Aug-2017.) (Proof shortened by Wolf Lammen, 13-Aug-2017.)
 |-  ( -.  ph  ->  A. x  -.  ph )   =>    |-  ( A. x ph  ->  ph )
 
Theoremcbvaliw 1641* Change bound variable. Uses only Tarski's FOL axiom schemes. Part of Lemma 7 of [KalishMontague] p. 86. (Contributed by NM, 19-Apr-2017.)
 |-  ( A. x ph  ->  A. y A. x ph )   &    |-  ( -.  ps  ->  A. x  -.  ps )   &    |-  ( x  =  y 
 ->  ( ph  ->  ps )
 )   =>    |-  ( A. x ph  ->  A. y ps )
 
Theoremcbvalivw 1642* Change bound variable. Uses only Tarski's FOL axiom schemes. Part of Lemma 7 of [KalishMontague] p. 86. (Contributed by NM, 9-Apr-2017.)
 |-  ( x  =  y 
 ->  ( ph  ->  ps )
 )   =>    |-  ( A. x ph  ->  A. y ps )
 
1.4.7  Axiom scheme ax-8 (Equality)
 
Axiomax-8 1643 Axiom of Equality. One of the equality and substitution axioms of predicate calculus with equality. This is similar to, but not quite, a transitive law for equality (proved later as equtr 1652). This axiom scheme is a sub-scheme of Axiom Scheme B8 of system S2 of [Tarski], p. 75, whose general form cannot be represented with our notation. Also appears as Axiom C7 of [Monk2] p. 105 and Axiom Scheme C8' in [Megill] p. 448 (p. 16 of the preprint).

The equality symbol was invented in 1527 by Robert Recorde. He chose a pair of parallel lines of the same length because "noe .2. thynges, can be moare equalle."

Note that this axiom is still valid even when any two or all three of  x,  y, and  z are replaced with the same variable since they do not have any distinct variable (Metamath's $d) restrictions. Because of this, we say that these three variables are "bundled" (a term coined by Raph Levien). (Contributed by NM, 5-Aug-1993.)

 |-  ( x  =  y 
 ->  ( x  =  z 
 ->  y  =  z
 ) )
 
Theoremequid 1644 Identity law for equality. Lemma 2 of [KalishMontague] p. 85. See also Lemma 6 of [Tarski] p. 68. (Contributed by NM, 1-Apr-2005.) (Revised by NM, 9-Apr-2017.)
 |-  x  =  x
 
Theoremnfequid 1645 Bound-variable hypothesis builder for  x  =  x. This theorem tells us that any variable, including  x, is effectively not free in  x  =  x, even though  x is technically free according to the traditional definition of free variable. (Contributed by NM, 13-Jan-2011.) (Revised by NM, 21-Aug-2017.)
 |- 
 F/ y  x  =  x
 
Theoremequcomi 1646 Commutative law for equality. Lemma 3 of [KalishMontague] p. 85. See also Lemma 7 of [Tarski] p. 69. (Contributed by NM, 5-Aug-1993.) (Revised by NM, 9-Apr-2017.)
 |-  ( x  =  y 
 ->  y  =  x )
 
Theoremequcom 1647 Commutative law for equality. (Contributed by NM, 20-Aug-1993.)
 |-  ( x  =  y  <-> 
 y  =  x )
 
Theoremequequ1 1648 An equivalence law for equality. (Contributed by NM, 5-Aug-1993.)
 |-  ( x  =  y 
 ->  ( x  =  z  <-> 
 y  =  z ) )
 
Theoremequequ2 1649 An equivalence law for equality. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Wolf Lammen, 4-Aug-2017.)
 |-  ( x  =  y 
 ->  ( z  =  x  <-> 
 z  =  y ) )
 
Theoremstdpc6 1650 One of the two equality axioms of standard predicate calculus, called reflexivity of equality. (The other one is stdpc7 1858.) Axiom 6 of [Mendelson] p. 95. Mendelson doesn't say why he prepended the redundant quantifier, but it was probably to be compatible with free logic (which is valid in the empty domain). (Contributed by NM, 16-Feb-2005.)
 |- 
 A. x  x  =  x
 
Theoremequcoms 1651 An inference commuting equality in antecedent. Used to eliminate the need for a syllogism. (Contributed by NM, 5-Aug-1993.)
 |-  ( x  =  y 
 ->  ph )   =>    |-  ( y  =  x 
 ->  ph )
 
Theoremequtr 1652 A transitive law for equality. (Contributed by NM, 23-Aug-1993.)
 |-  ( x  =  y 
 ->  ( y  =  z 
 ->  x  =  z
 ) )
 
Theoremequtrr 1653 A transitive law for equality. Lemma L17 in [Megill] p. 446 (p. 14 of the preprint). (Contributed by NM, 23-Aug-1993.)
 |-  ( x  =  y 
 ->  ( z  =  x 
 ->  z  =  y
 ) )
 
Theoremequtr2 1654 A transitive law for equality. (Contributed by NM, 12-Aug-1993.) (Proof shortened by Andrew Salmon, 25-May-2011.)
 |-  ( ( x  =  z  /\  y  =  z )  ->  x  =  y )
 
Theoremax12b 1655 Two equivalent ways of expressing ax-12 1866. See the comment for ax-12 1866. (Contributed by NM, 2-May-2017.) (Proof shortened by Wolf Lammen, 12-Aug-2017.)
 |-  ( ( -.  x  =  y  ->  ( y  =  z  ->  A. x  y  =  z )
 ) 
 <->  ( -.  x  =  y  ->  ( -.  x  =  z  ->  ( y  =  z  ->  A. x  y  =  z ) ) ) )
 
Theoremax12bOLD 1656 Obsolete version of ax12b 1655 as of 12-Aug-2017. (Contributed by NM, 2-May-2017.) (New usage is discouraged.)
 |-  ( ( -.  x  =  y  ->  ( y  =  z  ->  A. x  y  =  z )
 ) 
 <->  ( -.  x  =  y  ->  ( -.  x  =  z  ->  ( y  =  z  ->  A. x  y  =  z ) ) ) )
 
Theoremspfw 1657* Weak version of sp 1716. Uses only Tarski's FOL axiom schemes. Lemma 9 of [KalishMontague] p. 87. This may be the best we can do with minimal distinct variable conditions. TO DO: Do we need this theorem? If not, maybe it should be deleted. (Contributed by NM, 19-Apr-2017.)
 |-  ( -.  ps  ->  A. x  -.  ps )   &    |-  ( A. x ph  ->  A. y A. x ph )   &    |-  ( -.  ph  ->  A. y  -.  ph )   &    |-  ( x  =  y  ->  ( ph  <->  ps ) )   =>    |-  ( A. x ph  -> 
 ph )
 
TheoremspnfwOLD 1658 Weak version of sp 1716. Uses only Tarski's FOL axiom schemes. Obsolete version of spnfw 1640 as of 13-Aug-2017. (Contributed by NM, 1-Aug-2017.) (New usage is discouraged.)
 |-  ( -.  ph  ->  A. x  -.  ph )   =>    |-  ( A. x ph  ->  ph )
 
Theorem19.8w 1659 Weak version of 19.8a 1718. Uses only Tarski's FOL axiom schemes. (Contributed by NM, 1-Aug-2017.)
 |-  ( ph  ->  A. x ph )   =>    |-  ( ph  ->  E. x ph )
 
Theoremspw 1660* Weak version of specialization scheme sp 1716. Lemma 9 of [KalishMontague] p. 87. While it appears that sp 1716 in its general form does not follow from Tarski's FOL axiom schemes, from this theorem we can prove any instance of sp 1716 having no wff metavariables and mutually distinct set variables (see ax11wdemo 1697 for an example of the procedure to eliminate the hypothesis). Other approximations of sp 1716 are spfw 1657 (minimal distinct variable requirements), spnfw 1640 (when  x is not free in  -.  ph), spvw 1661 (when  x does not appear in  ph), sptruw 1669 (when  ph is true), and spfalw 1670 (when  ph is false). (Contributed by NM, 9-Apr-2017.)
 |-  ( x  =  y 
 ->  ( ph  <->  ps ) )   =>    |-  ( A. x ph 
 ->  ph )
 
Theoremspvw 1661* Version of sp 1716 when  x does not occur in  ph. This provides the other direction of ax-17 1603. Uses only Tarski's FOL axiom schemes. (Contributed by NM, 10-Apr-2017.)
 |-  ( A. x ph  -> 
 ph )
 
Theorem19.3v 1662* Special case of Theorem 19.3 of [Margaris] p. 89. (Contributed by NM, 1-Aug-2017.)
 |-  ( A. x ph  <->  ph )
 
Theorem19.9v 1663* Special case of Theorem 19.9 of [Margaris] p. 89. (Contributed by NM, 28-May-1995.) (Revised by NM, 1-Aug-2017.)
 |-  ( E. x ph  <->  ph )
 
Theoremexlimdv 1664* Deduction from Theorem 19.23 of [Margaris] p. 90. (Contributed by NM, 27-Apr-1994.)
 |-  ( ph  ->  ( ps  ->  ch ) )   =>    |-  ( ph  ->  ( E. x ps  ->  ch ) )
 
Theoremexlimddv 1665* Existential elimination rule of natural deduction. (Contributed by Mario Carneiro, 15-Jun-2016.)
 |-  ( ph  ->  E. x ps )   &    |-  ( ( ph  /\ 
 ps )  ->  ch )   =>    |-  ( ph  ->  ch )
 
Theoremexlimiv 1666* Inference from Theorem 19.23 of [Margaris] p. 90.

This inference, along with our many variants such as rexlimdv 2666, is used to implement a metatheorem called "Rule C" that is given in many logic textbooks. See, for example, Rule C in [Mendelson] p. 81, Rule C in [Margaris] p. 40, or Rule C in Hirst and Hirst's A Primer for Logic and Proof p. 59 (PDF p. 65) at http://www.mathsci.appstate.edu/~hirstjl/primer/hirst.pdf.

In informal proofs, the statement "Let  C be an element such that..." almost always means an implicit application of Rule C.

In essence, Rule C states that if we can prove that some element  x exists satisfying a wff, i.e.  E. x ph ( x ) where  ph ( x ) has  x free, then we can use  ph ( C ) as a hypothesis for the proof where  C is a new (ficticious) constant not appearing previously in the proof, nor in any axioms used, nor in the theorem to be proved. The purpose of Rule C is to get rid of the existential quantifier.

We cannot do this in Metamath directly. Instead, we use the original  ph (containing  x) as an antecedent for the main part of the proof. We eventually arrive at  ( ph  ->  ps ) where  ps is the theorem to be proved and does not contain  x. Then we apply exlimiv 1666 to arrive at  ( E. x ph  ->  ps ). Finally, we separately prove  E. x ph and detach it with modus ponens ax-mp 8 to arrive at the final theorem  ps. (Contributed by NM, 5-Aug-1993.)

 |-  ( ph  ->  ps )   =>    |-  ( E. x ph  ->  ps )
 
Theoremexlimivv 1667* Inference from Theorem 19.23 of [Margaris] p. 90. (Contributed by NM, 1-Aug-1995.)
 |-  ( ph  ->  ps )   =>    |-  ( E. x E. y ph  ->  ps )
 
Theoremexlimdvv 1668* Deduction from Theorem 19.23 of [Margaris] p. 90. (Contributed by NM, 31-Jul-1995.)
 |-  ( ph  ->  ( ps  ->  ch ) )   =>    |-  ( ph  ->  ( E. x E. y ps  ->  ch ) )
 
Theoremsptruw 1669 Version of sp 1716 when  ph is true. Uses only Tarski's FOL axiom schemes. (Contributed by NM, 23-Apr-1017.)
 |-  ph   =>    |-  ( A. x ph  -> 
 ph )
 
Theoremspfalw 1670 Version of sp 1716 when  ph is false. Uses only Tarski's FOL axiom schemes. (Contributed by NM, 23-Apr-1017.)
 |- 
 -.  ph   =>    |-  ( A. x ph  -> 
 ph )
 
Theorem19.2 1671 Theorem 19.2 of [Margaris] p. 89. Note: This proof is very different from Margaris' because we only have Tarski's FOL axiom schemes available at this point. See the later 19.2g 1780 for a more conventional proof. (Contributed by NM, 2-Aug-2017.)
 |-  ( A. x ph  ->  E. x ph )
 
Theorem19.39 1672 Theorem 19.39 of [Margaris] p. 90. (Contributed by NM, 5-Aug-1993.)
 |-  ( ( E. x ph 
 ->  E. x ps )  ->  E. x ( ph  ->  ps ) )
 
Theorem19.24 1673 Theorem 19.24 of [Margaris] p. 90. (Contributed by NM, 5-Aug-1993.)
 |-  ( ( A. x ph 
 ->  A. x ps )  ->  E. x ( ph  ->  ps ) )
 
Theorem19.34 1674 Theorem 19.34 of [Margaris] p. 90. (Contributed by NM, 5-Aug-1993.)
 |-  ( ( A. x ph 
 \/  E. x ps )  ->  E. x ( ph  \/  ps ) )
 
Theoremcbvalw 1675* Change bound variable. Uses only Tarski's FOL axiom schemes. (Contributed by NM, 9-Apr-2017.)
 |-  ( A. x ph  ->  A. y A. x ph )   &    |-  ( -.  ps  ->  A. x  -.  ps )   &    |-  ( A. y ps 
 ->  A. x A. y ps )   &    |-  ( -.  ph  ->  A. y  -.  ph )   &    |-  ( x  =  y 
 ->  ( ph  <->  ps ) )   =>    |-  ( A. x ph  <->  A. y ps )
 
Theoremcbvalvw 1676* Change bound variable. Uses only Tarski's FOL axiom schemes. (Contributed by NM, 9-Apr-2017.)
 |-  ( x  =  y 
 ->  ( ph  <->  ps ) )   =>    |-  ( A. x ph  <->  A. y ps )
 
Theoremcbvexvw 1677* Change bound variable. Uses only Tarski's FOL axiom schemes. (Contributed by NM, 19-Apr-2017.)
 |-  ( x  =  y 
 ->  ( ph  <->  ps ) )   =>    |-  ( E. x ph  <->  E. y ps )
 
Theoremalcomiw 1678* Weak version of alcom 1711. Uses only Tarski's FOL axiom schemes. (Contributed by NM, 10-Apr-2017.)
 |-  ( y  =  z 
 ->  ( ph  <->  ps ) )   =>    |-  ( A. x A. y ph  ->  A. y A. x ph )
 
Theoremhbn1fw 1679* Weak version of ax-6 1703 from which we can prove any ax-6 1703 instance not involving wff variables or bundling. Uses only Tarski's FOL axiom schemes. (Contributed by NM, 19-Apr-2017.)
 |-  ( A. x ph  ->  A. y A. x ph )   &    |-  ( -.  ps  ->  A. x  -.  ps )   &    |-  ( A. y ps 
 ->  A. x A. y ps )   &    |-  ( -.  ph  ->  A. y  -.  ph )   &    |-  ( -.  A. y ps  ->  A. x  -.  A. y ps )   &    |-  ( x  =  y  ->  ( ph  <->  ps ) )   =>    |-  ( -.  A. x ph 
 ->  A. x  -.  A. x ph )
 
Theoremhbn1w 1680* Weak version of hbn1 1704. Uses only Tarski's FOL axiom schemes. (Contributed by NM, 9-Apr-2017.)
 |-  ( x  =  y 
 ->  ( ph  <->  ps ) )   =>    |-  ( -.  A. x ph  ->  A. x  -.  A. x ph )
 
Theoremhba1w 1681* Weak version of hba1 1719. See comments for ax6w 1691. Uses only Tarski's FOL axiom schemes. (Contributed by NM, 9-Apr-2017.)
 |-  ( x  =  y 
 ->  ( ph  <->  ps ) )   =>    |-  ( A. x ph 
 ->  A. x A. x ph )
 
Theoremhbe1w 1682* Weak version of hbe1 1705. See comments for ax6w 1691. Uses only Tarski's FOL axiom schemes. (Contributed by NM, 19-Apr-2017.)
 |-  ( x  =  y 
 ->  ( ph  <->  ps ) )   =>    |-  ( E. x ph 
 ->  A. x E. x ph )
 
Theoremhbalw 1683* Weak version of hbal 1710. Uses only Tarski's FOL axiom schemes. Unlike hbal 1710, this theorem requires that  x and  y be distinct i.e. are not bundled. (Contributed by NM, 19-Apr-2017.)
 |-  ( x  =  z 
 ->  ( ph  <->  ps ) )   &    |-  ( ph  ->  A. x ph )   =>    |-  ( A. y ph  ->  A. x A. y ph )
 
1.4.8  Membership predicate
 
Syntaxwcel 1684 Extend wff definition to include the membership connective between classes.

For a general discussion of the theory of classes, see http://us.metamath.org/mpeuni/mmset.html#class.

(The purpose of introducing 
wff  A  e.  B here is to allow us to express i.e. "prove" the wel 1685 of predicate calculus in terms of the wceq 1623 of set theory, so that we don't "overload" the  e. connective with two syntax definitions. This is done to prevent ambiguity that would complicate some Metamath parsers. The class variables  A and  B are introduced temporarily for the purpose of this definition but otherwise not used in predicate calculus. See df-clab 2270 for more information on the set theory usage of wcel 1684.)

 wff  A  e.  B
 
Theoremwel 1685 Extend wff definition to include atomic formulas with the epsilon (membership) predicate. This is read " x is an element of  y," " x is a member of  y," " x belongs to  y," or " y contains  x." Note: The phrase " y includes  x " means " x is a subset of  y;" to use it also for  x  e.  y, as some authors occasionally do, is poor form and causes confusion, according to George Boolos (1992 lecture at MIT).

This syntactical construction introduces a binary non-logical predicate symbol  e. (epsilon) into our predicate calculus. We will eventually use it for the membership predicate of set theory, but that is irrelevant at this point: the predicate calculus axioms for  e. apply to any arbitrary binary predicate symbol. "Non-logical" means that the predicate is presumed to have additional properties beyond the realm of predicate calculus, although these additional properties are not specified by predicate calculus itself but rather by the axioms of a theory (in our case set theory) added to predicate calculus. "Binary" means that the predicate has two arguments.

(Instead of introducing wel 1685 as an axiomatic statement, as was done in an older version of this database, we introduce it by "proving" a special case of set theory's more general wcel 1684. This lets us avoid overloading the  e. connective, thus preventing ambiguity that would complicate certain Metamath parsers. However, logically wel 1685 is considered to be a primitive syntax, even though here it is artificially "derived" from wcel 1684. Note: To see the proof steps of this syntax proof, type "show proof wel /all" in the Metamath program.) (Contributed by NM, 24-Jan-2006.)

 wff  x  e.  y
 
1.4.9  Axiom schemes ax-13 (Left Membership Equality)
 
Axiomax-13 1686 Axiom of Left Membership Equality. One of the equality and substitution axioms for a non-logical predicate in our predicate calculus with equality. It substitutes equal variables into the left-hand side of the  e. binary predicate. This axiom scheme is a sub-scheme of Axiom Scheme B8 of system S2 of [Tarski], p. 75, whose general form cannot be represented with our notation. Also appears as Axiom scheme C12' in [Megill] p. 448 (p. 16 of the preprint). "Non-logical" means that the predicate is not a primitive of predicate calculus proper but instead is an extension to it. "Binary" means that the predicate has two arguments. In a system of predicate calculus with equality, like ours, equality is not usually considered to be a non-logical predicate. In systems of predicate calculus without equality, it typically would be. (Contributed by NM, 5-Aug-1993.)
 |-  ( x  =  y 
 ->  ( x  e.  z  ->  y  e.  z ) )
 
Theoremelequ1 1687 An identity law for the non-logical predicate. (Contributed by NM, 5-Aug-1993.)
 |-  ( x  =  y 
 ->  ( x  e.  z  <->  y  e.  z ) )
 
1.4.10  Axiom schemes ax-14 (Right Membership Equality)
 
Axiomax-14 1688 Axiom of Right Membership Equality. One of the equality and substitution axioms for a non-logical predicate in our predicate calculus with equality. It substitutes equal variables into the right-hand side of the  e. binary predicate. This axiom scheme is a sub-scheme of Axiom Scheme B8 of system S2 of [Tarski], p. 75, whose general form cannot be represented with our notation. Also appears as Axiom scheme C13' in [Megill] p. 448 (p. 16 of the preprint). (Contributed by NM, 5-Aug-1993.)
 |-  ( x  =  y 
 ->  ( z  e.  x  ->  z  e.  y ) )
 
Theoremelequ2 1689 An identity law for the non-logical predicate. (Contributed by NM, 5-Aug-1993.)
 |-  ( x  =  y 
 ->  ( z  e.  x  <->  z  e.  y ) )
 
1.4.11  Logical redundancy of ax-6 , ax-7 , ax-11 , ax-12

The orginal axiom schemes of Tarski's predicate calculus are ax-5 1544, ax-17 1603, ax9v 1636, ax-8 1643, ax-13 1686, and ax-14 1688, together with rule ax-gen 1533. See http://us.metamath.org/mpeuni/mmset.html#compare 1533. They are given as axiom schemes B4 through B8 in [KalishMontague] p. 81. These are shown to be logically complete by Theorem 1 of [KalishMontague] p. 85.

The axiom system of set.mm includes the auxiliary axiom schemes ax-6 1703, ax-7 1708, ax-12 1866, and ax-11 1715, which are not part of Tarski's axiom schemes. They are used (and we conjecture are required) to make our system "metalogically complete" i.e. able to prove directly all possible schemes with wff and set metavariables, bundled or not, whose object-language instances are valid. (ax-11 1715 has been proved to be required; see http://us.metamath.org/award2003.html#9a. Metalogical independence of the other three are open problems.)

(There are additional predicate calculus axiom schemes included in set.mm such as ax-4 2074, but they can all be proved as theorems from the above.)

Terminology: Two set (individual) metavariables are "bundled" in an axiom or theorem scheme when there is no distinct variable constraint ($d) imposed on them. (The term "bundled" is due to Raph Levien.) For example, the  x and  y in ax9 1889 are bundled, but they are not in ax9v 1636. We also say that a scheme is bundled when it has at least one pair of bundled set metavariables. If distinct variable conditions are added to all set metavariable pairs in a bundled scheme, we call that the "principal" instance of the bundled scheme. For example, ax9v 1636 is the principal instance of ax9 1889. Whenever a common variable is substituted for two or more bundled variables in an axiom or theorem scheme, we call the substitution instance "degenerate". For example, the instance  -.  A. x -.  x  =  x of ax9 1889 is degenerate. An advantage of bundling is ease of use since there are fewer distinct variable restrictions ($d) to be concerned with. There is also a small economy in being able to state principal and degenerate instances simultaneously. A disadvantage is that bundling may present difficulties in translations to other proof languages, which typically lack the concept (in part because their variables often represent the variables of the object language rather than metavariables ranging over them).

Because Tarski's axiom schemes are logically complete, they can be used to prove any object-language instance of ax-6 1703, ax-7 1708, ax-11 1715, and ax-12 1866 . "Translating" this to Metamath, it means that Tarski's axioms can prove any substitution instance of ax-6 1703, ax-7 1708, ax-11 1715, or ax-12 1866 in which (1) there are no wff metavariables and (2) all set metavariables are mutually distinct i.e. are not bundled. In effect this is mimicking the object language by pretending that each set metavariable is an object-language variable. (There may also be specific instances with wff metavariables and/or bundling that are directly provable from Tarski's axiom schemes, but it isn't guaranteed. Whether all of them are possible is part of the still open metalogical independence problem for our additional axiom schemes.)

It can be useful to see how this can be done, both to show that our additional schemes are valid metatheorems of Tarski's system and to be able to translate object language instances of our proofs into proofs that would work with a system using only Tarski's original schemes. In addition, it may (or may not) provide insight into the conjectured metalogical independence of our additional schemes.

The new theorem schemes ax6w 1691, ax7w 1692, ax11w 1695, and ax12w 1698 are derived using only Tarski's axiom schemes, showing that Tarski's schemes can be used to derive all substitution instances of ax-6 1703, ax-7 1708, ax-11 1715, and ax-12 1866 meeting conditions (1) and (2). (The "w" suffix stands for "weak version".) Each hypothesis of ax6w 1691, ax7w 1692, and ax11w 1695 is of the form  ( x  =  y  ->  ( ph  <->  ps ) ) where  ps is an auxiliary or "dummy" wff metavariable in which  x doesn't occur. We can show by induction on formula length that the hypotheses can be eliminated in all cases meeting conditions (1) and (2). The example ax11wdemo 1697 illustrates the techniques (equality theorems and bound variable renaming) used to achieve this.

We also show the degenerate instances for axioms with bundled variables in ax7dgen 1693, ax11dgen 1696, ax12dgen1 1699, ax12dgen2 1700, ax12dgen3 1701, and ax12dgen4 1702. (Their proofs are trivial but we include them to be thorough.) Combining the principal and degenerate cases outside of Metamath, we show that the bundled schemes ax-6 1703, ax-7 1708, ax-11 1715, and ax-12 1866 are schemes of Tarski's system, meaning that all object language instances they generate are theorems of Tarski's system.

It is interesting that Tarski used the bundled scheme ax-9 1635 in an older system, so it seems the main purpose of his later ax9v 1636 was just to show that the weaker unbundled form is sufficient rather than an aesthetic objection to bundled free and bound variables. Since we adopt the bundled ax-9 1635 as our official axiom, we show that the degenerate instance holds in ax9dgen 1690.

The case of sp 1716 is curious: originally an axiom of Tarski's system, it was proved redundant by Lemma 9 of [KalishMontague] p. 86. However, the proof is by induction on formula length, and the compact scheme form  A. x ph  ->  ph apparently cannot be proved directly from Tarski's other axioms. The best we can do seems to be spw 1660, again requiring substitution instances of  ph that meet conditions (1) and (2) above. Note that our direct proof sp 1716 requires ax-11 1715, which is not part of Tarski's system.

 
Theoremax9dgen 1690 Tarski's system uses the weaker ax9v 1636 instead of the bundled ax-9 1635, so here we show that the degenerate case of ax-9 1635 can be derived. (Contributed by NM, 23-Apr-2017.)
 |- 
 -.  A. x  -.  x  =  x
 
Theoremax6w 1691* Weak version of ax-6 1703 from which we can prove any ax-6 1703 instance not involving wff variables or bundling. Uses only Tarski's FOL axiom schemes. (Contributed by NM, 9-Apr-2017.)
 |-  ( x  =  y 
 ->  ( ph  <->  ps ) )   =>    |-  ( -.  A. x ph  ->  A. x  -.  A. x ph )
 
Theoremax7w 1692* Weak version of ax-7 1708 from which we can prove any ax-7 1708 instance not involving wff variables or bundling. Uses only Tarski's FOL axiom schemes. Unlike ax-7 1708, this theorem requires that  x and  y be distinct i.e. are not bundled. (Contributed by NM, 10-Apr-2017.)
 |-  ( y  =  z 
 ->  ( ph  <->  ps ) )   =>    |-  ( A. x A. y ph  ->  A. y A. x ph )
 
Theoremax7dgen 1693 Degenerate instance of ax-7 1708 where bundled variables  x and  y have a common substitution. Uses only Tarski's FOL axiom schemes. (Contributed by NM, 13-Apr-2017.)
 |-  ( A. x A. x ph  ->  A. x A. x ph )
 
Theoremax11wlem 1694* Lemma for weak version of ax-11 1715. Uses only Tarski's FOL axiom schemes. In some cases, this lemma may lead to shorter proofs than ax11w 1695. (Contributed by NM, 10-Apr-2017.)
 |-  ( x  =  y 
 ->  ( ph  <->  ps ) )   =>    |-  ( x  =  y  ->  ( ph  ->  A. x ( x  =  y  ->  ph )
 ) )
 
Theoremax11w 1695* Weak version of ax-11 1715 from which we can prove any ax-11 1715 instance not involving wff variables or bundling. Uses only Tarski's FOL axiom schemes. An instance of the first hypothesis will normally require that  x and  y be distinct (unless  x does not occur in  ph). (Contributed by NM, 10-Apr-2017.)
 |-  ( x  =  y 
 ->  ( ph  <->  ps ) )   &    |-  (
 y  =  z  ->  ( ph  <->  ch ) )   =>    |-  ( x  =  y  ->  ( A. y ph  ->  A. x ( x  =  y  ->  ph ) ) )
 
Theoremax11dgen 1696 Degenerate instance of ax-11 1715 where bundled variables  x and  y have a common substitution. Uses only Tarski's FOL axiom schemes. (Contributed by NM, 13-Apr-2017.)
 |-  ( x  =  x 
 ->  ( A. x ph  ->  A. x ( x  =  x  ->  ph )
 ) )
 
Theoremax11wdemo 1697* Example of an application of ax11w 1695 that results in an instance of ax-11 1715 for a contrived formula with mixed free and bound variables,  ( x  e.  y  /\  A. x
z  e.  x  /\  A. y A. z y  e.  x ), in place of  ph. The proof illustrates bound variable renaming with cbvalvw 1676 to obtain fresh variables to avoid distinct variable clashes. Uses only Tarski's FOL axiom schemes. (Contributed by NM, 14-Apr-2017.)
 |-  ( x  =  y 
 ->  ( A. y ( x  e.  y  /\  A. x  z  e.  x  /\  A. y A. z  y  e.  x )  ->  A. x ( x  =  y  ->  ( x  e.  y  /\  A. x  z  e.  x  /\  A. y A. z  y  e.  x )
 ) ) )
 
Theoremax12w 1698* Weak version (principal instance) of ax-12 1866 not involving bundling. Uses only Tarski's FOL axiom schemes. The proof is trivial but is included to complete the set ax6w 1691, ax7w 1692, and ax11w 1695. (Contributed by NM, 10-Apr-2017.)
 |-  ( -.  x  =  y  ->  ( y  =  z  ->  A. x  y  =  z )
 )
 
Theoremax12dgen1 1699 Degenerate instance of ax-12 1866 where bundled variables  x and  y have a common substitution. Uses only Tarski's FOL axiom schemes. (Contributed by NM, 13-Apr-2017.)
 |-  ( -.  x  =  x  ->  ( x  =  z  ->  A. x  x  =  z )
 )
 
Theoremax12dgen2 1700 Degenerate instance of ax-12 1866 where bundled variables  x and  z have a common substitution. Uses only Tarski's FOL axiom schemes. (Contributed by NM, 13-Apr-2017.)
 |-  ( -.  x  =  y  ->  ( y  =  x  ->  A. x  y  =  x )
 )
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