Home Metamath Proof ExplorerTheorem List (p. 17 of 329) < Previous  Next > Browser slow? Try the Unicode version. Mirrors  >  Metamath Home Page  >  MPE Home Page  >  Theorem List Contents  >  Recent Proofs       This page: Page List

 Color key: Metamath Proof Explorer (1-22426) Hilbert Space Explorer (22427-23949) Users' Mathboxes (23950-32836)

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.)

Theoremexbidh 1602 Formula-building rule for existential quantifier (deduction rule). (Contributed by NM, 5-Aug-1993.)

Theoremexsimpl 1603 Simplification of an existentially quantified conjunction. (Contributed by Rodolfo Medina, 25-Sep-2010.) (Proof shortened by Andrew Salmon, 29-Jun-2011.)

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.)

Theorem19.26-2 1605 Theorem 19.26 of [Margaris] p. 90 with two quantifiers. (Contributed by NM, 3-Feb-2005.)

Theorem19.26-3an 1606 Theorem 19.26 of [Margaris] p. 90 with triple conjunction. (Contributed by NM, 13-Sep-2011.)

Theorem19.29 1607 Theorem 19.29 of [Margaris] p. 90. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Andrew Salmon, 13-May-2011.)

Theorem19.29r 1608 Variation of Theorem 19.29 of [Margaris] p. 90. (Contributed by NM, 18-Aug-1993.)

Theorem19.29r2 1609 Variation of Theorem 19.29 of [Margaris] p. 90 with double quantification. (Contributed by NM, 3-Feb-2005.)

Theorem19.29x 1610 Variation of Theorem 19.29 of [Margaris] p. 90 with mixed quantification. (Contributed by NM, 11-Feb-2005.)

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.)

Theorem19.35i 1612 Inference from Theorem 19.35 of [Margaris] p. 90. (Contributed by NM, 5-Aug-1993.)

Theorem19.35ri 1613 Inference from Theorem 19.35 of [Margaris] p. 90. (Contributed by NM, 5-Aug-1993.)

Theorem19.25 1614 Theorem 19.25 of [Margaris] p. 90. (Contributed by NM, 5-Aug-1993.)

Theorem19.30 1615 Theorem 19.30 of [Margaris] p. 90. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Andrew Salmon, 25-May-2011.)

Theorem19.43 1616 Theorem 19.43 of [Margaris] p. 90. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Wolf Lammen, 27-Jun-2014.)

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.)

Theorem19.33 1618 Theorem 19.33 of [Margaris] p. 90. (Contributed by NM, 5-Aug-1993.)

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.)

Theorem19.40 1620 Theorem 19.40 of [Margaris] p. 90. (Contributed by NM, 5-Aug-1993.)

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.)

Theoremalbiim 1622 Split a biconditional and distribute quantifier. (Contributed by NM, 18-Aug-1993.)

Theorem2albiim 1623 Split a biconditional and distribute 2 quantifiers. (Contributed by NM, 3-Feb-2005.)

Theoremexintrbi 1624 Add/remove a conjunct in the scope of an existential quantifier. (Contributed by Raph Levien, 3-Jul-2006.)

Theoremexintr 1625 Introduce a conjunct in the scope of an existential quantifier. (Contributed by NM, 11-Aug-1993.)

Theoremalsyl 1626 Theorem *10.3 in [WhiteheadRussell] p. 150. (Contributed by Andrew Salmon, 8-Jun-2011.)

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 does not occur in , i.e. does not depend on in any way, then we can add the quantifier to with no further assumptions. By sp 1764, we can also remove the quantifier (unconditionally). (Contributed by NM, 5-Aug-1993.)

Theorema17d 1628* ax-17 1627 with antecedent. Useful in proofs of deduction versions of bound-variable hypothesis builders. (Contributed by NM, 1-Mar-2013.)

Theoremax17e 1629* A rephrasing of ax-17 1627 using the existential quantifier. (Contributed by Wolf Lammen, 4-Dec-2017.)

Theoremnfv 1630* If is not present in , then is not free in . (Contributed by Mario Carneiro, 11-Aug-2016.)

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.)

Theoremalimdv 1632* Deduction from Theorem 19.20 of [Margaris] p. 90. (Contributed by NM, 3-Apr-1994.)

Theoremeximdv 1633* Deduction from Theorem 19.22 of [Margaris] p. 90. (Contributed by NM, 27-Apr-1994.)

Theorem2alimdv 1634* Deduction from Theorem 19.22 of [Margaris] p. 90. (Contributed by NM, 27-Apr-2004.)

Theorem2eximdv 1635* Deduction from Theorem 19.22 of [Margaris] p. 90. (Contributed by NM, 3-Aug-1995.)

Theoremalbidv 1636* Formula-building rule for universal quantifier (deduction rule). (Contributed by NM, 5-Aug-1993.)

Theoremexbidv 1637* Formula-building rule for existential quantifier (deduction rule). (Contributed by NM, 5-Aug-1993.)

Theorem2albidv 1638* Formula-building rule for 2 universal quantifiers (deduction rule). (Contributed by NM, 4-Mar-1997.)

Theorem2exbidv 1639* Formula-building rule for 2 existential quantifiers (deduction rule). (Contributed by NM, 1-May-1995.)

Theorem3exbidv 1640* Formula-building rule for 3 existential quantifiers (deduction rule). (Contributed by NM, 1-May-1995.)

Theorem4exbidv 1641* Formula-building rule for 4 existential quantifiers (deduction rule). (Contributed by NM, 3-Aug-1995.)

Theoremalrimiv 1642* Inference from Theorem 19.21 of [Margaris] p. 90. (Contributed by NM, 5-Aug-1993.)

Theoremalrimivv 1643* Inference from Theorem 19.21 of [Margaris] p. 90. (Contributed by NM, 31-Jul-1995.)

Theoremalrimdv 1644* Deduction from Theorem 19.21 of [Margaris] p. 90. (Contributed by NM, 10-Feb-1997.)

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 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 exists satisfying a wff, i.e. where has free, then we can use as a hypothesis for the proof where 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 (containing ) as an antecedent for the main part of the proof. We eventually arrive at where is the theorem to be proved and does not contain . Then we apply exlimiv 1645 to arrive at . Finally, we separately prove and detach it with modus ponens ax-mp 5 to arrive at the final theorem . (Contributed by NM, 5-Aug-1993.) (Revised by Wolf Lammen to remove dependency on ax-9 and ax-8, 4-Dec-2017.)

Theoremexlimivv 1646* Inference from Theorem 19.23 of [Margaris] p. 90. (Contributed by NM, 1-Aug-1995.)

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.)

Theoremexlimdvv 1648* Deduction from Theorem 19.23 of [Margaris] p. 90. (Contributed by NM, 31-Jul-1995.)

Theoremexlimddv 1649* Existential elimination rule of natural deduction. (Contributed by Mario Carneiro, 15-Jun-2016.)

Theoremnfdv 1650* Apply the definition of not-free in a context. (Contributed by Mario Carneiro, 11-Aug-2016.)

Theorem2ax17 1651* Quantification of two variables over a formula in which they do not occur. (Contributed by Alan Sare, 12-Apr-2011.)

1.4.5  Equality predicate; define substitution

Syntaxcv 1652 This syntax construction states that a variable , 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 is a class by cab 2424. Since (when is distinct from ) we have by cvjust 2433, we can argue that the syntax " " can be viewed as an abbreviation for " ". 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 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.)

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 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 could be the of either weq 1654 or wceq 1653, although mathematically it makes no difference. The class variables and 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.)

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.)

Theoremequs3 1655 Lemma used in proofs of substitution properties. (Contributed by NM, 5-Aug-1993.)

Theoremspeimfw 1656 Specialization, with additional weakening to allow bundling of and . Uses only Tarski's FOL axiom schemes. (Contributed by NM, 23-Apr-2017.) (Proof shortened by Wolf Lammen, 5-Aug-2017.)

Theoremspimfw 1657 Specialization, with additional weakening to allow bundling of and . Uses only Tarski's FOL axiom schemes. (Contributed by NM, 23-Apr-1017.) (Proof shortened by Wolf Lammen, 7-Aug-2017.)

Theoremax11i 1658 Inference that has ax-11 1762 (without ) 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.)

Syntaxwsb 1659 Extend wff definition to include proper substitution (read "the wff that results when is properly substituted for in wff "). (Contributed by NM, 24-Jan-2006.)

Definitiondf-sb 1660 Define proper substitution. Remark 9.1 in [Megill] p. 447 (p. 15 of the preprint). For our notation, we use to mean "the wff that results from the proper substitution of for in the wff ." We can also use 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, " is the wff that results when is properly substituted for in ." For example, if the original is , then is , from which we obtain that is . So what exactly does 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 and are replaced with the same variable, as sbid 1948 shows. We achieve this by having 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 and 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 . (Contributed by NM, 5-Aug-1993.)

Theoremsbequ2 1661 An equality theorem for substitution. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Wolf Lammen, 25-Feb-2018.)

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.)

Theoremsb1 1663 One direction of a simplified definition of substitution. (Contributed by NM, 5-Aug-1993.)

Theoremspsbe 1664 A specialization theorem. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Wolf Lammen, 3-May-2018.)

Theoremsbimi 1665 Infer substitution into antecedent and consequent of an implication. (Contributed by NM, 25-Jun-1998.)

Theoremsbbii 1666 Infer substitution into both sides of a logical equivalence. (Contributed by NM, 5-Aug-1993.)

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 and 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.)

Theoremax9v 1668* Axiom B7 of [Tarski] p. 75, which requires that and 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 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 and . 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.)

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.)

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.)

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.)

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.)

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.)

Theorem19.39 1674 Theorem 19.39 of [Margaris] p. 90. (Contributed by NM, 5-Aug-1993.)

Theorem19.24 1675 Theorem 19.24 of [Margaris] p. 90. (Contributed by NM, 5-Aug-1993.)

Theorem19.34 1676 Theorem 19.34 of [Margaris] p. 90. (Contributed by NM, 5-Aug-1993.)

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.)

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.)

Theoremspvw 1679* Version of sp 1764 when does not occur in . 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.)

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.)

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.)

Theoremspimvw 1682* Specialization. Lemma 8 of [KalishMontague] p. 87. Uses only Tarski's FOL axiom schemes. (Contributed by NM, 9-Apr-2017.)

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.)

Theoremsptruw 1684 Version of sp 1764 when is true. Uses only Tarski's FOL axiom schemes. (Contributed by NM, 23-Apr-1017.)

Theoremspfalw 1685 Version of sp 1764 when is false. Uses only Tarski's FOL axiom schemes. (Contributed by NM, 23-Apr-1017.) (Proof shortened by Wolf Lammen, 25-Dec-2017.)

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.)

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.)

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 , , and 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.)

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.)

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.)

Theoremnfequid 1691 Bound-variable hypothesis builder for . This theorem tells us that any variable, including , is effectively not free in , even though is technically free according to the traditional definition of free variable. (Contributed by NM, 13-Jan-2011.) (Revised by NM, 21-Aug-2017.)

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.)

Theoremequcom 1693 Commutative law for equality. (Contributed by NM, 20-Aug-1993.)

Theoremequcoms 1694 An inference commuting equality in antecedent. Used to eliminate the need for a syllogism. (Contributed by NM, 5-Aug-1993.)

Theoremequtr 1695 A transitive law for equality. (Contributed by NM, 23-Aug-1993.)

Theoremequtrr 1696 A transitive law for equality. Lemma L17 in [Megill] p. 446 (p. 14 of the preprint). (Contributed by NM, 23-Aug-1993.)

Theoremequequ1 1697 An equivalence law for equality. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Wolf Lammen, 10-Dec-2017.)

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.)

Theoremequequ2 1699 An equivalence law for equality. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Wolf Lammen, 4-Aug-2017.)

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.)

Page List
Jump to page: Contents  1 1-100 2 101-200 3 201-300 4 301-400 5 401-500 6 501-600 7 601-700 8 701-800 9 801-900 10 901-1000 11 1001-1100 12 1101-1200 13 1201-1300 14 1301-1400 15 1401-1500 16 1501-1600 17 1601-1700 18 1701-1800 19 1801-1900 20 1901-2000 21 2001-2100 22 2101-2200 23 2201-2300 24 2301-2400 25 2401-2500 26 2501-2600 27 2601-2700 28 2701-2800 29 2801-2900 30 2901-3000 31 3001-3100 32 3101-3200 33 3201-3300 34 3301-3400 35 3401-3500 36 3501-3600 37 3601-3700 38 3701-3800 39 3801-3900 40 3901-4000 41 4001-4100 42 4101-4200 43 4201-4300 44 4301-4400 45 4401-4500 46 4501-4600 47 4601-4700 48 4701-4800 49 4801-4900 50 4901-5000 51 5001-5100 52 5101-5200 53 5201-5300 54 5301-5400 55 5401-5500 56 5501-5600 57 5601-5700 58 5701-5800 59 5801-5900 60 5901-6000 61 6001-6100 62 6101-6200 63 6201-6300 64 6301-6400 65 6401-6500 66 6501-6600 67 6601-6700 68 6701-6800 69 6801-6900 70 6901-7000 71 7001-7100 72 7101-7200 73 7201-7300 74 7301-7400 75 7401-7500 76 7501-7600 77 7601-7700 78 7701-7800 79 7801-7900 80 7901-8000 81 8001-8100 82 8101-8200 83 8201-8300 84 8301-8400 85 8401-8500 86 8501-8600 87 8601-8700 88 8701-8800 89 8801-8900 90 8901-9000 91 9001-9100 92 9101-9200 93 9201-9300 94 9301-9400 95 9401-9500 96 9501-9600 97 9601-9700 98 9701-9800 99 9801-9900 100 9901-10000 101 10001-10100 102 10101-10200 103 10201-10300 104 10301-10400 105 10401-10500 106 10501-10600 107 10601-10700 108 10701-10800 109 10801-10900 110 10901-11000 111 11001-11100 112 11101-11200 113 11201-11300 114 11301-11400 115 11401-11500 116 11501-11600 117 11601-11700 118 11701-11800 119 11801-11900 120 11901-12000 121 12001-12100 122 12101-12200 123 12201-12300 124 12301-12400 125 12401-12500 126 12501-12600 127 12601-12700 128 12701-12800 129 12801-12900 130 12901-13000 131 13001-13100 132 13101-13200 133 13201-13300 134 13301-13400 135 13401-13500 136 13501-13600 137 13601-13700 138 13701-13800 139 13801-13900 140 13901-14000 141 14001-14100 142 14101-14200 143 14201-14300 144 14301-14400 145 14401-14500 146 14501-14600 147 14601-14700 148 14701-14800 149 14801-14900 150 14901-15000 151 15001-15100 152 15101-15200 153 15201-15300 154 15301-15400 155 15401-15500 156 15501-15600 157 15601-15700 158 15701-15800 159 15801-15900 160 15901-16000 161 16001-16100 162 16101-16200 163 16201-16300 164 16301-16400 165 16401-16500 166 16501-16600 167 16601-16700 168 16701-16800 169 16801-16900 170 16901-17000 171 17001-17100 172 17101-17200 173 17201-17300 174 17301-17400 175 17401-17500 176 17501-17600 177 17601-17700 178 17701-17800 179 17801-17900 180 17901-18000 181 18001-18100 182 18101-18200 183 18201-18300 184 18301-18400 185 18401-18500 186 18501-18600 187 18601-18700 188 18701-18800 189 18801-18900 190 18901-19000 191 19001-19100 192 19101-19200 193 19201-19300 194 19301-19400 195 19401-19500 196 19501-19600 197 19601-19700 198 19701-19800 199 19801-19900 200 19901-20000 201 20001-20100 202 20101-20200 203 20201-20300 204 20301-20400 205 20401-20500 206 20501-20600 207 20601-20700 208 20701-20800 209 20801-20900 210 20901-21000 211 21001-21100 212 21101-21200 213 21201-21300 214 21301-21400 215 21401-21500 216 21501-21600 217 21601-21700 218 21701-21800 219 21801-21900 220 21901-22000 221 22001-22100 222 22101-22200 223 22201-22300 224 22301-22400 225 22401-22500 226 22501-22600 227 22601-22700 228 22701-22800 229 22801-22900 230 22901-23000 231 23001-23100 232 23101-23200 233 23201-23300 234 23301-23400 235 23401-23500 236 23501-23600 237 23601-23700 238 23701-23800 239 23801-23900 240 23901-24000 241 24001-24100 242 24101-24200 243 24201-24300 244 24301-24400 245 24401-24500 246 24501-24600 247 24601-24700 248 24701-24800 249 24801-24900 250 24901-25000 251 25001-25100 252 25101-25200 253 25201-25300 254 25301-25400 255 25401-25500 256 25501-25600 257 25601-25700 258 25701-25800 259 25801-25900 260 25901-26000 261 26001-26100 262 26101-26200 263 26201-26300 264 26301-26400 265 26401-26500 266 26501-26600 267 26601-26700 268 26701-26800 269 26801-26900 270 26901-27000 271 27001-27100 272 27101-27200 273 27201-27300 274 27301-27400 275 27401-27500 276 27501-27600 277 27601-27700 278 27701-27800 279 27801-27900 280 27901-28000 281 28001-28100 282 28101-28200 283 28201-28300 284 28301-28400 285 28401-28500 286 28501-28600 287 28601-28700 288 28701-28800 289 28801-28900 290 28901-29000 291 29001-29100 292 29101-29200 293 29201-29300 294 29301-29400 295 29401-29500 296 29501-29600 297 29601-29700 298 29701-29800 299 29801-29900 300 29901-30000 301 30001-30100 302 30101-30200 303 30201-30300 304 30301-30400 305 30401-30500 306 30501-30600 307 30601-30700 308 30701-30800 309 30801-30900 310 30901-31000 311 31001-31100 312 31101-31200 313 31201-31300 314 31301-31400 315 31401-31500 316 31501-31600 317 31601-31700 318 31701-31800 319 31801-31900 320 31901-32000 321 32001-32100 322 32101-32200 323 32201-32300 324 32301-32400 325 32401-32500 326 32501-32600 327 32601-32700 328 32701-32800 329 32801-32836
 Copyright terms: Public domain < Previous  Next >