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Theorem List for Metamath Proof Explorer - 1601-1700   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremalbidh 1601 Formula-building rule for universal quantifier (deduction rule). (Contributed by NM, 5-Aug-1993.)
 |-  ( ph  ->  A. x ph )   &    |-  ( ph  ->  ( ps  <->  ch ) )   =>    |-  ( ph  ->  (
 A. x ps  <->  A. x ch )
 )
 
Theoremexbidh 1602 Formula-building rule for existential quantifier (deduction rule). (Contributed by NM, 5-Aug-1993.)
 |-  ( ph  ->  A. x ph )   &    |-  ( ph  ->  ( ps  <->  ch ) )   =>    |-  ( ph  ->  ( E. x ps  <->  E. x ch )
 )
 
Theoremexsimpl 1603 Simplification of an existentially quantified conjunction. (Contributed by Rodolfo Medina, 25-Sep-2010.) (Proof shortened by Andrew Salmon, 29-Jun-2011.)
 |-  ( E. x (
 ph  /\  ps )  ->  E. x ph )
 
Theorem19.26 1604 Theorem 19.26 of [Margaris] p. 90. Also Theorem *10.22 of [WhiteheadRussell] p. 147. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Wolf Lammen, 4-Jul-2014.)
 |-  ( A. x (
 ph  /\  ps )  <->  (
 A. x ph  /\  A. x ps ) )
 
Theorem19.26-2 1605 Theorem 19.26 of [Margaris] p. 90 with two quantifiers. (Contributed by NM, 3-Feb-2005.)
 |-  ( A. x A. y ( ph  /\  ps ) 
 <->  ( A. x A. y ph  /\  A. x A. y ps ) )
 
Theorem19.26-3an 1606 Theorem 19.26 of [Margaris] p. 90 with triple conjunction. (Contributed by NM, 13-Sep-2011.)
 |-  ( A. x (
 ph  /\  ps  /\  ch ) 
 <->  ( A. x ph  /\ 
 A. x ps  /\  A. x ch ) )
 
Theorem19.29 1607 Theorem 19.29 of [Margaris] p. 90. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Andrew Salmon, 13-May-2011.)
 |-  ( ( A. x ph 
 /\  E. x ps )  ->  E. x ( ph  /\ 
 ps ) )
 
Theorem19.29r 1608 Variation of Theorem 19.29 of [Margaris] p. 90. (Contributed by NM, 18-Aug-1993.)
 |-  ( ( E. x ph 
 /\  A. x ps )  ->  E. x ( ph  /\ 
 ps ) )
 
Theorem19.29r2 1609 Variation of Theorem 19.29 of [Margaris] p. 90 with double quantification. (Contributed by NM, 3-Feb-2005.)
 |-  ( ( E. x E. y ph  /\  A. x A. y ps )  ->  E. x E. y
 ( ph  /\  ps )
 )
 
Theorem19.29x 1610 Variation of Theorem 19.29 of [Margaris] p. 90 with mixed quantification. (Contributed by NM, 11-Feb-2005.)
 |-  ( ( E. x A. y ph  /\  A. x E. y ps )  ->  E. x E. y
 ( ph  /\  ps )
 )
 
Theorem19.35 1611 Theorem 19.35 of [Margaris] p. 90. This theorem is useful for moving an implication (in the form of the right-hand side) into the scope of a single existential quantifier. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Wolf Lammen, 27-Jun-2014.)
 |-  ( E. x (
 ph  ->  ps )  <->  ( A. x ph 
 ->  E. x ps )
 )
 
Theorem19.35i 1612 Inference from Theorem 19.35 of [Margaris] p. 90. (Contributed by NM, 5-Aug-1993.)
 |- 
 E. x ( ph  ->  ps )   =>    |-  ( A. x ph  ->  E. x ps )
 
Theorem19.35ri 1613 Inference from Theorem 19.35 of [Margaris] p. 90. (Contributed by NM, 5-Aug-1993.)
 |-  ( A. x ph  ->  E. x ps )   =>    |-  E. x ( ph  ->  ps )
 
Theorem19.25 1614 Theorem 19.25 of [Margaris] p. 90. (Contributed by NM, 5-Aug-1993.)
 |-  ( A. y E. x ( ph  ->  ps )  ->  ( E. y A. x ph  ->  E. y E. x ps ) )
 
Theorem19.30 1615 Theorem 19.30 of [Margaris] p. 90. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Andrew Salmon, 25-May-2011.)
 |-  ( A. x (
 ph  \/  ps )  ->  ( A. x ph  \/  E. x ps )
 )
 
Theorem19.43 1616 Theorem 19.43 of [Margaris] p. 90. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Wolf Lammen, 27-Jun-2014.)
 |-  ( E. x (
 ph  \/  ps )  <->  ( E. x ph  \/  E. x ps ) )
 
Theorem19.43OLD 1617 Obsolete proof of 19.43 1616 as of 3-May-2017. Leave this in for the example on the mmrecent.html page. (Contributed by NM, 5-Aug-1993.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  ( E. x (
 ph  \/  ps )  <->  ( E. x ph  \/  E. x ps ) )
 
Theorem19.33 1618 Theorem 19.33 of [Margaris] p. 90. (Contributed by NM, 5-Aug-1993.)
 |-  ( ( A. x ph 
 \/  A. x ps )  ->  A. x ( ph  \/  ps ) )
 
Theorem19.33b 1619 The antecedent provides a condition implying the converse of 19.33 1618. Compare Theorem 19.33 of [Margaris] p. 90. (Contributed by NM, 27-Mar-2004.) (Proof shortened by Andrew Salmon, 25-May-2011.) (Proof shortened by Wolf Lammen, 5-Jul-2014.)
 |-  ( -.  ( E. x ph  /\  E. x ps )  ->  ( A. x ( ph  \/  ps )  <->  ( A. x ph 
 \/  A. x ps )
 ) )
 
Theorem19.40 1620 Theorem 19.40 of [Margaris] p. 90. (Contributed by NM, 5-Aug-1993.)
 |-  ( E. x (
 ph  /\  ps )  ->  ( E. x ph  /\ 
 E. x ps )
 )
 
Theorem19.40-2 1621 Theorem *11.42 in [WhiteheadRussell] p. 163. Theorem 19.40 of [Margaris] p. 90 with 2 quantifiers. (Contributed by Andrew Salmon, 24-May-2011.)
 |-  ( E. x E. y ( ph  /\  ps )  ->  ( E. x E. y ph  /\  E. x E. y ps )
 )
 
Theoremalbiim 1622 Split a biconditional and distribute quantifier. (Contributed by NM, 18-Aug-1993.)
 |-  ( A. x (
 ph 
 <->  ps )  <->  ( A. x ( ph  ->  ps )  /\  A. x ( ps 
 ->  ph ) ) )
 
Theorem2albiim 1623 Split a biconditional and distribute 2 quantifiers. (Contributed by NM, 3-Feb-2005.)
 |-  ( A. x A. y ( ph  <->  ps )  <->  ( A. x A. y ( ph  ->  ps )  /\  A. x A. y ( ps  ->  ph ) ) )
 
Theoremexintrbi 1624 Add/remove a conjunct in the scope of an existential quantifier. (Contributed by Raph Levien, 3-Jul-2006.)
 |-  ( A. x (
 ph  ->  ps )  ->  ( E. x ph  <->  E. x ( ph  /\ 
 ps ) ) )
 
Theoremexintr 1625 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 1626 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 1627* 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 2236 about the logical redundancy of ax-17 1627 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 1764, we can also remove the quantifier (unconditionally). (Contributed by NM, 5-Aug-1993.)

 |-  ( ph  ->  A. x ph )
 
Theorema17d 1628* ax-17 1627 with antecedent. Useful in proofs of deduction versions of bound-variable hypothesis builders. (Contributed by NM, 1-Mar-2013.)
 |-  ( ph  ->  ( ps  ->  A. x ps )
 )
 
Theoremax17e 1629* A rephrasing of ax-17 1627 using the existential quantifier. (Contributed by Wolf Lammen, 4-Dec-2017.)
 |-  ( E. x ph  -> 
 ph )
 
Theoremnfv 1630* 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 1631* nfv 1630 with antecedent. Useful in proofs of deduction versions of bound-variable hypothesis builders such as nfimd 1828. (Contributed by Mario Carneiro, 6-Oct-2016.)
 |-  ( ph  ->  F/ x ps )
 
Theoremalimdv 1632* 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 1633* 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 1634* 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 1635* 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 1636* 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 1637* 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 1638* 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 1639* 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 1640* 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 1641* 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 1642* Inference from Theorem 19.21 of [Margaris] p. 90. (Contributed by NM, 5-Aug-1993.)
 |-  ( ph  ->  ps )   =>    |-  ( ph  ->  A. x ps )
 
Theoremalrimivv 1643* Inference from Theorem 19.21 of [Margaris] p. 90. (Contributed by NM, 31-Jul-1995.)
 |-  ( ph  ->  ps )   =>    |-  ( ph  ->  A. x A. y ps )
 
Theoremalrimdv 1644* Deduction from Theorem 19.21 of [Margaris] p. 90. (Contributed by NM, 10-Feb-1997.)
 |-  ( ph  ->  ( ps  ->  ch ) )   =>    |-  ( ph  ->  ( ps  ->  A. x ch ) )
 
Theoremexlimiv 1645* Inference from Theorem 19.23 of [Margaris] p. 90.

This inference, along with our many variants such as rexlimdv 2831, 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 1645 to arrive at  ( E. x ph  ->  ps ). Finally, we separately prove  E. x ph and detach it with modus ponens ax-mp 5 to arrive at the final theorem  ps. (Contributed by NM, 5-Aug-1993.) (Revised by Wolf Lammen to remove dependency on ax-9 and ax-8, 4-Dec-2017.)

 |-  ( ph  ->  ps )   =>    |-  ( E. x ph  ->  ps )
 
Theoremexlimivv 1646* Inference from Theorem 19.23 of [Margaris] p. 90. (Contributed by NM, 1-Aug-1995.)
 |-  ( ph  ->  ps )   =>    |-  ( E. x E. y ph  ->  ps )
 
Theoremexlimdv 1647* Deduction from Theorem 19.23 of [Margaris] p. 90. Revised to remove dependency on ax-9 1667 and ax-8 1688. (Contributed by NM, 27-Apr-1994.) (Revised by Wolf Lammen, 4-Dec-2017.)
 |-  ( ph  ->  ( ps  ->  ch ) )   =>    |-  ( ph  ->  ( E. x ps  ->  ch ) )
 
Theoremexlimdvv 1648* 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 ) )
 
Theoremexlimddv 1649* Existential elimination rule of natural deduction. (Contributed by Mario Carneiro, 15-Jun-2016.)
 |-  ( ph  ->  E. x ps )   &    |-  ( ( ph  /\ 
 ps )  ->  ch )   =>    |-  ( ph  ->  ch )
 
Theoremnfdv 1650* 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 1651* 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 1652 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 2424. Since (when  y is distinct from  x) we have  x  =  { y  |  y  e.  x } by cvjust 2433, we can argue 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 1652 as a "type conversion" from a set variable to a class variable, keep in mind that cv 1652 is intrinsically no different from any other class-building syntax such as cab 2424, cun 3320, or c0 3630.

For a general discussion of the theory of classes and the role of cv 1652, 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 1654 of predicate calculus from the wceq 1653 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 1653 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 1654 of predicate calculus in terms of the wceq 1653 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 1654 or wceq 1653, 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 2431 for more information on the set theory usage of wceq 1653.)

 wff  A  =  B
 
Theoremweq 1654 Extend wff definition to include atomic formulas using the equality predicate.

(Instead of introducing weq 1654 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 1653. This lets us avoid overloading the  = connective, thus preventing ambiguity that would complicate certain Metamath parsers. However, logically weq 1654 is considered to be a primitive syntax, even though here it is artificially "derived" from wceq 1653. 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 1655 Lemma used in proofs of substitution properties. (Contributed by NM, 5-Aug-1993.)
 |-  ( E. x ( x  =  y  /\  ph )  <->  -.  A. x ( x  =  y  ->  -.  ph ) )
 
Theoremspeimfw 1656 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 1657 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 1658 Inference that has ax-11 1762 (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 1659 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 1660 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 2092.

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 2113, sbcom2 2192 and sbid2v 2202).

Note that our definition is valid even when  x and  y are replaced with the same variable, as sbid 1948 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 2200 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 2111. When  x and  y are distinct, we can express proper substitution with the simpler expressions of sb5 2178 and sb6 2177.

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 1661 An equality theorem for substitution. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Wolf Lammen, 25-Feb-2018.)
 |-  ( x  =  y 
 ->  ( [ y  /  x ] ph  ->  ph )
 )
 
Theoremsbequ2OLD 1662 Obsolete proof of sbequ2 1661 as of 25-Feb-2018. (Contributed by NM, 5-Aug-1993.) (New usage is discouraged.) (Proof modification is discouraged.)
 |-  ( x  =  y 
 ->  ( [ y  /  x ] ph  ->  ph )
 )
 
Theoremsb1 1663 One direction of a simplified definition of substitution. (Contributed by NM, 5-Aug-1993.)
 |-  ( [ y  /  x ] ph  ->  E. x ( x  =  y  /\  ph ) )
 
Theoremspsbe 1664 A specialization theorem. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Wolf Lammen, 3-May-2018.)
 |-  ( [ y  /  x ] ph  ->  E. x ph )
 
Theoremsbimi 1665 Infer substitution into antecedent and consequent of an implication. (Contributed by NM, 25-Jun-1998.)
 |-  ( ph  ->  ps )   =>    |-  ( [ y  /  x ] ph  ->  [ y  /  x ] ps )
 
Theoremsbbii 1666 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 1667 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 1955 and ax9from9o 2227. A more convenient form of this axiom is a9e 1953, 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 1667 can be proved from the weaker version ax9v 1668 requiring that the variables be distinct; see theorem ax9 1954.

ax-9 1667 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 4337.

Except by ax9v 1668, this axiom should not be referenced directly. Instead, use theorem ax9 1954. (Contributed by NM, 5-Aug-1993.) (New usage is discouraged.)

 |- 
 -.  A. x  -.  x  =  y
 
Theoremax9v 1668* 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 1667 by adding a distinct variable restriction. From here on, ax-9 1667 should not be referenced directly by any other proof, so that theorem ax9 1954 will show that we can recover ax-9 1667 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 1668 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 1667.

When possible, use of this theorem rather than ax9 1954 is preferred since its derivation from axioms is much shorter. (Contributed by NM, 7-Aug-2015.)

 |- 
 -.  A. x  -.  x  =  y
 
Theorema9ev 1669* At least one individual exists. Weaker version of a9e 1953. When possible, use of this theorem rather than a9e 1953 is preferred since its derivation from axioms is much shorter. (Contributed by NM, 3-Aug-2017.)
 |- 
 E. x  x  =  y
 
Theoremexiftru 1670 A companion rule to ax-gen, valid only if an individual exists. Unlike ax-9 1667, it does not require equality on its interface. Some fundamental theorems of predicate logic can be proven from ax-gen 1556, ax-5 1567 and this theorem alone, not requiring ax-8 1688 or excessive distinct variable conditions. (Contributed by Wolf Lammen, 12-Nov-2017.) (Proof shortened by Wolf Lammen, 9-Dec-2017.)
 |-  ph   =>    |- 
 E. x ph
 
TheoremexiftruOLD 1671 Obsolete proof of exiftru 1670 as of 9-Dec-2017. (Contributed by Wolf Lammen, 12-Nov-2017.) (New usage is discouraged.) (Proof modification is discouraged.)
 |-  ph   =>    |- 
 E. x ph
 
Theorem19.2 1672 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 1774 for a more conventional proof. Revised to remove dependency on ax-8 1688. (Contributed by NM, 2-Aug-2017.) (Revised by Wolf Lammen, 4-Dec-2017.)
 |-  ( A. x ph  ->  E. x ph )
 
Theorem19.8w 1673 Weak version of 19.8a 1763. Uses only Tarski's FOL axiom schemes. (Contributed by NM, 1-Aug-2017.) (Proof shortened by Wolf Lammen, 4-Dec-2017.)
 |-  ( ph  ->  A. x ph )   =>    |-  ( ph  ->  E. x ph )
 
Theorem19.39 1674 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 1675 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 1676 Theorem 19.34 of [Margaris] p. 90. (Contributed by NM, 5-Aug-1993.)
 |-  ( ( A. x ph 
 \/  E. x ps )  ->  E. x ( ph  \/  ps ) )
 
Theorem19.9v 1677* Special case of Theorem 19.9 of [Margaris] p. 89. Revised to remove dependency on ax-8 1688. (Contributed by NM, 28-May-1995.) (Revised by NM, 1-Aug-2017.) (Revised by Wolf Lammen, 4-Dec-2017.)
 |-  ( E. x ph  <->  ph )
 
Theorem19.3v 1678* Special case of Theorem 19.3 of [Margaris] p. 89. Revised to remove dependency on ax-8 1688. (Contributed by NM, 1-Aug-2017.) (Revised by Wolf Lammen, 4-Dec-2017.)
 |-  ( A. x ph  <->  ph )
 
Theoremspvw 1679* Version of sp 1764 when  x does not occur in  ph. This provides the other direction of ax-17 1627. Uses only Tarski's FOL axiom schemes. (Contributed by NM, 10-Apr-2017.) (Proof shortened by Wolf Lammen, 4-Dec-2017.)
 |-  ( A. x ph  -> 
 ph )
 
Theoremspimeh 1680* Existential introduction, using implicit substitution. Compare Lemma 14 of [Tarski] p. 70. (Contributed by NM, 7-Aug-1994.) (Proof shortened by Wolf Lammen, 10-Dec-2017.)
 |-  ( ph  ->  A. x ph )   &    |-  ( x  =  z  ->  ( ph  ->  ps ) )   =>    |-  ( ph  ->  E. x ps )
 
Theoremspimw 1681* 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 1682* 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 1683 Weak version of sp 1764. 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 )
 
Theoremsptruw 1684 Version of sp 1764 when  ph is true. Uses only Tarski's FOL axiom schemes. (Contributed by NM, 23-Apr-1017.)
 |-  ph   =>    |-  ( A. x ph  -> 
 ph )
 
Theoremspfalw 1685 Version of sp 1764 when  ph is false. Uses only Tarski's FOL axiom schemes. (Contributed by NM, 23-Apr-1017.) (Proof shortened by Wolf Lammen, 25-Dec-2017.)
 |- 
 -.  ph   =>    |-  ( A. x ph  -> 
 ph )
 
Theoremcbvaliw 1686* 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 1687* 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 1688 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 1695). 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 1689 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.) (Proof shortened by Wolf Lammen, 5-Feb-2018.)
 |-  x  =  x
 
TheoremequidOLD 1690 Obsolete proof of equid 1689 as of 9-Dec-2017. (Contributed by NM, 1-Apr-2005.) (New usage is discouraged.) (Proof modification is discouraged.)
 |-  x  =  x
 
Theoremnfequid 1691 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 1692 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 1693 Commutative law for equality. (Contributed by NM, 20-Aug-1993.)
 |-  ( x  =  y  <-> 
 y  =  x )
 
Theoremequcoms 1694 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 1695 A transitive law for equality. (Contributed by NM, 23-Aug-1993.)
 |-  ( x  =  y 
 ->  ( y  =  z 
 ->  x  =  z
 ) )
 
Theoremequtrr 1696 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
 ) )
 
Theoremequequ1 1697 An equivalence law for equality. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Wolf Lammen, 10-Dec-2017.)
 |-  ( x  =  y 
 ->  ( x  =  z  <-> 
 y  =  z ) )
 
Theoremequequ1OLD 1698 Obsolete version of equequ1 1697 as of 12-Nov-2017. (Contributed by NM, 5-Aug-1993.) (New usage is discouraged.) (Proof modification is discouraged.)
 |-  ( x  =  y 
 ->  ( x  =  z  <-> 
 y  =  z ) )
 
Theoremequequ2 1699 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 1700 One of the two equality axioms of standard predicate calculus, called reflexivity of equality. (The other one is stdpc7 1943.) 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
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