P. 19 of Theorem List - Metamath Proof Explorer

Theorem List for Metamath Proof Explorer - 1801-1900   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	19.9v 1801*	Version of 19.9 1943 with a dv condition, requiring fewer axioms. Any formula can be existentially quantified using a variable which it does not contain. See also 19.3v 1802. (Contributed by NM, 28-May-1995.) Remove dependency on ax-7 1839. (Revised by Wolf Lammen, 4-Dec-2017.)
|- ( E. x ph <-> ph )
Theorem	19.3v 1802*	Version of 19.3 1939 with a dv condition, requiring fewer axioms. Any formula can be universally quantified using a variable which it does not contain. See also 19.9v 1801. (Contributed by Anthony Hart, 13-Sep-2011.) Remove dependency on ax-7 1839. (Revised by Wolf Lammen, 4-Dec-2017.)
|- ( A. x ph <-> ph )
Theorem	spvw 1803*	Version of sp 1910 when x does not occur in ph. Converse of ax-5 1748. 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 )
Theorem	19.39 1804	Theorem 19.39 of [Margaris] p. 90. (Contributed by NM, 12-Mar-1993.)
|- ( ( E. x ph -> E. x ps ) -> E. x ( ph -> ps ) )
Theorem	19.24 1805	Theorem 19.24 of [Margaris] p. 90. (Contributed by NM, 12-Mar-1993.)
|- ( ( A. x ph -> A. x ps ) -> E. x ( ph -> ps ) )
Theorem	19.34 1806	Theorem 19.34 of [Margaris] p. 90. (Contributed by NM, 12-Mar-1993.)
|- ( ( A. x ph \/ E. x ps ) -> E. x ( ph \/ ps ) )
Theorem	19.23v 1807*	Version of 19.23 1966 with a dv condition instead of a non-freeness hypothesis. (Contributed by NM, 28-Jun-1998.) Reduce dependencies on axioms. (Revised by Wolf Lammen, 11-Jan-2020.)
|- ( A. x ( ph -> ps ) <-> ( E. x ph -> ps ) )
Theorem	19.23vv 1808*	Theorem 19.23v 1807 extended to two variables. (Contributed by NM, 10-Aug-2004.)
|- ( A. x A. y ( ph -> ps ) <-> ( E. x E. y ph -> ps ) )
Theorem	19.36v 1809*	Version of 19.36 2019 with a dv condition instead of a non-freeness hypothesis. (Contributed by NM, 18-Aug-1993.) Reduce dependencies on axioms. (Revised by Wolf Lammen, 17-Jan-2020.)
|- ( E. x ( ph -> ps ) <-> ( A. x ph -> ps ) )
Theorem	19.36iv 1810*	Inference associated with 19.36v 1809. Version of 19.36i 2020 with a dv condition. (Contributed by NM, 5-Aug-1993.) Reduce dependencies on axioms. (Revised by Wolf Lammen, 17-Jan-2020.)
|- E. x ( ph -> ps )   =>    |- ( A. x ph -> ps )
Theorem	pm11.53v 1811*	Version of pm11.53 2038 with a dv condition, requiring fewer axioms. (Contributed by BJ, 7-Mar-2020.)
|- ( A. x A. y ( ph -> ps ) <-> ( E. x ph -> A. y ps ) )
Theorem	19.12vvv 1812*	Version of 19.12vv 2041 with a dv condition, requiring fewer axioms. See also 19.12 2006. (Contributed by BJ, 18-Mar-2020.)
|- ( E. x A. y ( ph -> ps ) <-> A. y E. x ( ph -> ps ) )
Theorem	19.27v 1813*	Version of 19.27 1979 with a dv condition, requiring fewer axioms. (Contributed by NM, 3-Jun-2004.)
|- ( A. x ( ph /\ ps ) <-> ( A. x ph /\ ps ) )
Theorem	19.28v 1814*	Version of 19.28 1980 with a dv condition, requiring fewer axioms. (Contributed by NM, 25-Mar-2004.)
|- ( A. x ( ph /\ ps ) <-> ( ph /\ A. x ps ) )
Theorem	19.37v 1815*	Version of 19.37 2021 with a dv condition, requiring fewer axioms. (Contributed by NM, 21-Jun-1993.)
|- ( E. x ( ph -> ps ) <-> ( ph -> E. x ps ) )
Theorem	19.37iv 1816*	Inference associated with 19.37v 1815. (Contributed by NM, 5-Aug-1993.)
|- E. x ( ph -> ps )   =>    |- ( ph -> E. x ps )
Theorem	19.44v 1817*	Version of 19.44 2024 with a dv condition, requiring fewer axioms (Contributed by NM, 12-Mar-1993.)
|- ( E. x ( ph \/ ps ) <-> ( E. x ph \/ ps ) )
Theorem	19.45v 1818*	Version of 19.45 2025 with a dv condition, requiring fewer axioms (Contributed by NM, 12-Mar-1993.)
|- ( E. x ( ph \/ ps ) <-> ( ph \/ E. x ps ) )
Theorem	19.41v 1819*	Version of 19.41 2026 with a dv condition, requiring fewer axioms. (Contributed by NM, 21-Jun-1993.)
|- ( E. x ( ph /\ ps ) <-> ( E. x ph /\ ps ) )
Theorem	19.41vv 1820*	Version of 19.41 2026 with two quantifiers and a dv condition requiring fewer axioms. (Contributed by NM, 30-Apr-1995.)
|- ( E. x E. y ( ph /\ ps ) <-> ( E. x E. y ph /\ ps ) )
Theorem	19.41vvv 1821*	Version of 19.41 2026 with three quantifiers and a dv condition requiring fewer axioms. (Contributed by NM, 30-Apr-1995.)
|- ( E. x E. y E. z ( ph /\ ps ) <-> ( E. x E. y E. z ph /\ ps ) )
Theorem	19.41vvvv 1822*	Version of 19.41 2026 with four quantifiers and a dv condition requiring fewer axioms. (Contributed by FL, 14-Jul-2007.)
|- ( E. w E. x E. y E. z ( ph /\ ps ) <-> ( E. w E. x E. y E. z ph /\ ps ) )
Theorem	19.42v 1823*	Version of 19.42 2027 with a dv condition requiring fewer axioms. (Contributed by NM, 21-Jun-1993.)
|- ( E. x ( ph /\ ps ) <-> ( ph /\ E. x ps ) )
Theorem	exdistr 1824*	Distribution of existential quantifiers. (Contributed by NM, 9-Mar-1995.)
|- ( E. x E. y ( ph /\ ps ) <-> E. x ( ph /\ E. y ps ) )
Theorem	19.42vv 1825*	Version of 19.42 2027 with two quantifiers and a dv condition requiring fewer axioms. (Contributed by NM, 16-Mar-1995.)
|- ( E. x E. y ( ph /\ ps ) <-> ( ph /\ E. x E. y ps ) )
Theorem	19.42vvv 1826*	Version of 19.42 2027 with three quantifiers and a dv condition requiring fewer axioms. (Contributed by NM, 21-Sep-2011.)
|- ( E. x E. y E. z ( ph /\ ps ) <-> ( ph /\ E. x E. y E. z ps ) )
Theorem	exdistr2 1827*	Distribution of existential quantifiers. (Contributed by NM, 17-Mar-1995.)
|- ( E. x E. y E. z ( ph /\ ps ) <-> E. x ( ph /\ E. y E. z ps ) )
Theorem	3exdistr 1828*	Distribution of existential quantifiers in a triple conjunction. (Contributed by NM, 9-Mar-1995.) (Proof shortened by Andrew Salmon, 25-May-2011.)
|- ( E. x E. y E. z ( ph /\ ps /\ ch ) <-> E. x ( ph /\ E. y ( ps /\ E. z ch ) ) )
Theorem	4exdistr 1829*	Distribution of existential quantifiers in a quadruple conjunction. (Contributed by NM, 9-Mar-1995.) (Proof shortened by Wolf Lammen, 20-Jan-2018.)
|- ( E. x E. y E. z E. w ( ( ph /\ ps ) /\ ( ch /\ th ) ) <-> E. x ( ph /\ E. y ( ps /\ E. z ( ch /\ E. w th ) ) ) )
Theorem	spimeh 1830*	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 = y -> ( ph -> ps ) )   =>    |- ( ph -> E. x ps )
Theorem	spimw 1831*	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 )
Theorem	spimvw 1832*	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 )
Theorem	spnfw 1833	Weak version of sp 1910. 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 )
Theorem	spfalw 1834	Version of sp 1910 when ph is false. Uses only Tarski's FOL axiom schemes. (Contributed by NM, 23-Apr-2017.) (Proof shortened by Wolf Lammen, 25-Dec-2017.)
|- -. ph   =>    |- ( A. x ph -> ph )
Theorem	equs4v 1835*	Version of equs4 2088 with a dv condition, which requires fewer axioms. (Contributed by BJ, 31-May-2019.)
|- ( A. x ( x = y -> ph ) -> E. x ( x = y /\ ph ) )
Theorem	equsalvw 1836*	Version of equsal 2089 with two dv conditions, which does not require ax-7 1839, ax-10 1887, ax-12 1905, ax-13 2053. (Contributed by BJ, 31-May-2019.)
|- ( x = y -> ( ph <-> ps ) )   =>    |- ( A. x ( x = y -> ph ) <-> ps )
Theorem	cbvaliw 1837*	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 )
Theorem	cbvalivw 1838*	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 )
1.4.8  Axiom scheme ax-7 (Equality)
Axiom	ax-7 1839	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 1846). 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, 10-Jan-1993.)
|- ( x = y -> ( x = z -> y = z ) )
Theorem	equid 1840	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.) (Proof shortened by Wolf Lammen, 22-Aug-2020.)
|- x = x
Theorem	equidOLD 1841	Obsolete proof of equid 1840 as of 22-Aug-2020. (Contributed by NM, 1-Apr-2005.) (Revised by NM, 9-Apr-2017.) (Proof shortened by Wolf Lammen, 5-Feb-2018.) (Proof modification is discouraged.) (New usage is discouraged.)
|- x = x
Theorem	nfequid 1842	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
Theorem	equcomi 1843	Commutative law for equality. Lemma 3 of [KalishMontague] p. 85. See also Lemma 7 of [Tarski] p. 69. (Contributed by NM, 10-Jan-1993.) (Revised by NM, 9-Apr-2017.)
|- ( x = y -> y = x )
Theorem	equcom 1844	Commutative law for equality. (Contributed by NM, 20-Aug-1993.)
|- ( x = y <-> y = x )
Theorem	equcoms 1845	An inference commuting equality in antecedent. Used to eliminate the need for a syllogism. (Contributed by NM, 10-Jan-1993.)
|- ( x = y -> ph )   =>    |- ( y = x -> ph )
Theorem	equtr 1846	A transitive law for equality. (Contributed by NM, 23-Aug-1993.)
|- ( x = y -> ( y = z -> x = z ) )
Theorem	equtrr 1847	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 ) )
Theorem	equequ1 1848	An equivalence law for equality. (Contributed by NM, 1-Aug-1993.) (Proof shortened by Wolf Lammen, 10-Dec-2017.)
|- ( x = y -> ( x = z <-> y = z ) )
Theorem	equequ2 1849	An equivalence law for equality. (Contributed by NM, 21-Jun-1993.) (Proof shortened by Wolf Lammen, 4-Aug-2017.)
|- ( x = y -> ( z = x <-> z = y ) )
Theorem	stdpc6 1850	One of the two equality axioms of standard predicate calculus, called reflexivity of equality. (The other one is stdpc7 1851.) 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
Theorem	stdpc7 1851	One of the two equality axioms of standard predicate calculus, called substitutivity of equality. (The other one is stdpc6 1850.) Translated to traditional notation, it can be read: " x = y -> ( ph ( x , x ) -> ph ( x , y ) ), provided that y is free for x in ph ( x , x )." Axiom 7 of [Mendelson] p. 95. (Contributed by NM, 15-Feb-2005.)
|- ( x = y -> ( [ x / y ] ph -> ph ) )
Theorem	equtr2 1852	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 )
Theorem	equviniv 1853*	A specialized version of equvini 2142 with a distinct variable restriction. (Contributed by Wolf Lammen, 8-Sep-2018.)
|- ( x = y -> E. z ( x = z /\ y = z ) )
Theorem	equvin 1854*	A variable introduction law for equality. Lemma 15 of [Monk2] p. 109. (Contributed by NM, 9-Jan-1993.) Remove dependencies on ax-10 1887, ax-13 2053. (Revised by Wolf Lammen, 10-Jun-2019.)
|- ( x = y <-> E. z ( x = z /\ z = y ) )
Theorem	ax13b 1855	An equivalence used to show two ways of expressing ax-13 2053. See the comment for ax-13 2053. (Contributed by NM, 2-May-2017.) (Proof shortened by Wolf Lammen, 26-Feb-2018.) (Revised by BJ, 15-Sep-2020.)
|- ( ( -. x = y -> ( y = z -> ph ) ) <-> ( -. x = y -> ( -. x = z -> ( y = z -> ph ) ) ) )
Theorem	spfw 1856*	Weak version of sp 1910. 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. (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 )
Theorem	spw 1857*	Weak version of the specialization scheme sp 1910. Lemma 9 of [KalishMontague] p. 87. While it appears that sp 1910 in its general form does not follow from Tarski's FOL axiom schemes, from this theorem we can prove any instance of sp 1910 having mutually distinct setvar variables and no wff metavariables (see ax12wdemo 1881 for an example of the procedure to eliminate the hypothesis). Other approximations of sp 1910 are spfw 1856 (minimal distinct variable requirements), spnfw 1833 (when x is not free in -. ph), spvw 1803 (when x does not appear in ph), sptruw 1677 (when ph is true), and spfalw 1834 (when ph is false). (Contributed by NM, 9-Apr-2017.) (Proof shortened by Wolf Lammen, 27-Feb-2018.)
|- ( x = y -> ( ph <-> ps ) )   =>    |- ( A. x ph -> ph )
Theorem	cbvalw 1858*	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 )
Theorem	cbvalvw 1859*	Change bound variable. Uses only Tarski's FOL axiom schemes. (Contributed by NM, 9-Apr-2017.) (Proof shortened by Wolf Lammen, 28-Feb-2018.)
|- ( x = y -> ( ph <-> ps ) )   =>    |- ( A. x ph <-> A. y ps )
Theorem	cbvexvw 1860*	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 )
Theorem	alcomiw 1861*	Weak version of alcom 1895. 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 )
Theorem	hbn1fw 1862*	Weak version of ax-10 1887 from which we can prove any ax-10 1887 instance not involving wff variables or bundling. Uses only Tarski's FOL axiom schemes. (Contributed by NM, 19-Apr-2017.) (Proof shortened by Wolf Lammen, 28-Feb-2018.)
|- ( 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 )
Theorem	hbn1w 1863*	Weak version of hbn1 1888. 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 )
Theorem	hba1w 1864*	Weak version of hba1 1951. See comments for ax10w 1875. 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 )
Theorem	hbe1w 1865*	Weak version of hbe1 1889. See comments for ax10w 1875. 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 )
Theorem	hbalw 1866*	Weak version of hbal 1894. Uses only Tarski's FOL axiom schemes. Unlike hbal 1894, 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 )
Theorem	cbvaev 1867*	Change bound variable in an equality with a dv condition. (Contributed by NM, 22-Jul-2015.) (Revised by BJ, 18-Jun-2019.)
|- ( A. x x = w -> A. y y = w )
1.4.9  Membership predicate
Syntax	wcel 1868	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 1869 of predicate calculus in terms of the wcel 1868 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 2408 for more information on the set theory usage of wcel 1868.)
wff A e. B
Theorem	wel 1869	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 1869 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 1868. This lets us avoid overloading the e. connective, thus preventing ambiguity that would complicate certain Metamath parsers. However, logically wel 1869 is considered to be a primitive syntax, even though here it is artificially "derived" from wcel 1868. 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
1.4.10  Axiom scheme ax-8 (Left Equality for Binary Predicate)
Axiom	ax-8 1870	Axiom of Left Equality for Binary Predicate. 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 an arbitrary binary predicate e., which we will use for the set membership relation when set theory is introduced. 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, 30-Jun-1993.)
|- ( x = y -> ( x e. z -> y e. z ) )
Theorem	elequ1 1871	An identity law for the non-logical predicate. (Contributed by NM, 30-Jun-1993.)
|- ( x = y -> ( x e. z <-> y e. z ) )
1.4.11  Axiom scheme ax-9 (Right Equality for Binary Predicate)
Axiom	ax-9 1872	Axiom of Right Equality for Binary Predicate. 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 an arbitrary binary predicate e., which we will use for the set membership relation when set theory is introduced. 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, 21-Jun-1993.)
|- ( x = y -> ( z e. x -> z e. y ) )
Theorem	elequ2 1873	An identity law for the non-logical predicate. (Contributed by NM, 21-Jun-1993.)
|- ( x = y -> ( z e. x <-> z e. y ) )
1.4.12  Logical redundancy of ax-10 , ax-11 , ax-12 , ax-13 The original axiom schemes of Tarski's predicate calculus are ax-4 1678, ax-5 1748, ax6v 1795, ax-7 1839, ax-8 1870, and ax-9 1872, together with rule ax-gen 1665. See http://us.metamath.org/mpeuni/mmset.html#compare 1665. 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-10 1887, ax-11 1892, ax-12 1905, and ax-13 2053, which are not part of Tarski's axiom schemes. Each object-language instance of them is provable from Tarski's axioms, so they are logically redundant. However, they are conjectured not to be provable directly as schemes from Tarski's axiom schemes using only Metamath's direct substitution rule. They are used to make our system "metalogically complete" i.e. able to prove directly all possible schemes with wff and setvar variables, bundled or not, whose object-language instances are valid. (ax-12 1905 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-c5 32374, but they can all be proved as theorems from the above.) Terminology: Two setvar (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 ax-6 1794 are bundled, but they are not in ax6v 1795. We also say that a scheme is bundled when it has at least one pair of bundled setvar variables. If distinct variable conditions are added to all setvar variable pairs in a bundled scheme, we call that the "principal" instance of the bundled scheme. For example, ax6v 1795 is the principal instance of ax-6 1794. 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 ax-6 1794 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-10 1887, ax-11 1892, ax-12 1905, and ax-13 2053. "Translating" this to Metamath, it means that Tarski's axioms can prove any substitution instance of ax-10 1887, ax-11 1892, ax-12 1905, or ax-13 2053 in which (1) there are no wff metavariables and (2) all setvar variables are mutually distinct i.e. are not bundled. In effect this is mimicking the object language by pretending that each setvar variable 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 theorem schemes ax10w 1875, ax11w 1876, ax12w 1879, and ax13w 1882 are derived using only Tarski's axiom schemes, showing that Tarski's schemes can be used to derive all substitution instances of ax-10 1887, ax-11 1892, ax-12 1905, and ax-13 2053 meeting Conditions (1) and (2). (The "w" suffix stands for "weak version".) Each hypothesis of ax10w 1875, ax11w 1876, and ax12w 1879 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 ax12wdemo 1881 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 ax11dgen 1877, ax12dgen 1880, ax13dgen1 1883, ax13dgen2 1884, ax13dgen3 1885, and ax13dgen4 1886. (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-10 1887, ax-11 1892, ax-12 1905, and ax-13 2053 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-6 1794 in an older system, so it seems the main purpose of his later ax6v 1795 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-6 1794 as our official axiom, we show that the degenerate instance holds in ax6dgen 1874. (Recall that in set.mm, the only statement referencing ax-6 1794 is ax6v 1795.) The case of sp 1910 is curious: originally an axiom of Tarski's system, it was proved logically redundant by Lemma 9 of [KalishMontague] p. 86. However, the proof is by induction on formula length, and the scheme form A. x ph -> ph apparently cannot be proved directly from Tarski's other axiom schemes. The best we can do seems to be spw 1857, again requiring substitution instances of ph that meet Conditions (1) and (2) above. Note that our direct proof sp 1910 requires ax-12 1905, which is not part of Tarski's system.
Theorem	ax6dgen 1874	Tarski's system uses the weaker ax6v 1795 instead of the bundled ax-6 1794, so here we show that the degenerate case of ax-6 1794 can be derived. Even though ax-6 1794 is in the list of axioms used, recall that in set.mm, the only statement referencing ax-6 1794 is ax6v 1795. We later re-derive from ax6v 1795 the bundled form as ax6 2057 with the help of the auxiliary axiom schemes. (Contributed by NM, 23-Apr-2017.)
|- -. A. x -. x = x
Theorem	ax10w 1875*	Weak version of ax-10 1887 from which we can prove any ax-10 1887 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 )
Theorem	ax11w 1876*	Weak version of ax-11 1892 from which we can prove any ax-11 1892 instance not involving wff variables or bundling. Uses only Tarski's FOL axiom schemes. Unlike ax-11 1892, 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 )
Theorem	ax11dgen 1877	Degenerate instance of ax-11 1892 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 )
Theorem	ax12wlem 1878*	Lemma for weak version of ax-12 1905. Uses only Tarski's FOL axiom schemes. In some cases, this lemma may lead to shorter proofs than ax12w 1879. (Contributed by NM, 10-Apr-2017.)
|- ( x = y -> ( ph <-> ps ) )   =>    |- ( x = y -> ( ph -> A. x ( x = y -> ph ) ) )
Theorem	ax12w 1879*	Weak version of ax-12 1905 from which we can prove any ax-12 1905 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). For an example of how the hypotheses can be eliminated when we substitute an expression without wff variables for ph, see ax12wdemo 1881. (Contributed by NM, 10-Apr-2017.)
|- ( x = y -> ( ph <-> ps ) )   &    |- ( y = z -> ( ph <-> ch ) )   =>    |- ( x = y -> ( A. y ph -> A. x ( x = y -> ph ) ) )
Theorem	ax12dgen 1880	Degenerate instance of ax-12 1905 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 ) ) )
Theorem	ax12wdemo 1881*	Example of an application of ax12w 1879 that results in an instance of ax-12 1905 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 1859 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 ) ) ) )
Theorem	ax13w 1882*	Weak version (principal instance) of ax-13 2053. (Because y and z don't need to be distinct, this actually bundles the principal instance and the degenerate instance ( -. x = y -> ( y = y -> A. x y = y ) ).) Uses only Tarski's FOL axiom schemes. The proof is trivial but is included to complete the set ax10w 1875, ax11w 1876, and ax12w 1879. (Contributed by NM, 10-Apr-2017.)
|- ( -. x = y -> ( y = z -> A. x y = z ) )
Theorem	ax13dgen1 1883	Degenerate instance of ax-13 2053 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 ) )
Theorem	ax13dgen2 1884	Degenerate instance of ax-13 2053 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 ) )
Theorem	ax13dgen3 1885	Degenerate instance of ax-13 2053 where bundled variables y and z have a common substitution. Uses only Tarski's FOL axiom schemes. (Contributed by NM, 13-Apr-2017.)
|- ( -. x = y -> ( y = y -> A. x y = y ) )
Theorem	ax13dgen4 1886	Degenerate instance of ax-13 2053 where bundled variables x, y, and z have a common substitution. Uses only Tarski's FOL axiom schemes . (Contributed by NM, 13-Apr-2017.)
|- ( -. x = x -> ( x = x -> A. x x = x ) )
1.5  Predicate calculus with equality: Auxiliary axiom schemes (4 schemes) In this section we introduce four additional schemes ax-10 1887, ax-11 1892, ax-12 1905, and ax-13 2053 that are not part of Tarski's system but can be proved (outside of Metamath) as theorem schemes of Tarski's system. These are needed to give our system the property of "metalogical completeness," which means that we can prove (with Metamath) all possible theorem schemes expressible in our language of wff metavariables ranging over object-language wffs, and setvar variables ranging over object-language individual variables. To show that these schemes are valid metatheorems of Tarski's system S2, above we proved from Tarski's system theorems ax10w 1875, ax11w 1876, ax12w 1879, and ax13w 1882, which show that any object-language instance of these schemes (emulated by having no wff metavariables and requiring all setvar variables to be mutually distinct) can be proved using only the schemes in Tarski's system S2. An open problem is to show that these four additional schemes are mutually metalogically independent and metalogically independent from Tarski's. So far, independence of ax-12 1905 from all others has been shown, and independence of Tarski's ax-6 1794 from all others has been shown; see items 9a and 11 on http://us.metamath.org/award2003.html.
1.5.1  Axiom scheme ax-10 (Quantified Negation)
Axiom	ax-10 1887	Axiom of Quantified Negation. Axiom C5-2 of [Monk2] p. 113. This axiom scheme is logically redundant (see ax10w 1875) but is used as an auxiliary axiom scheme to achieve metalogical completeness. It means that x is not free in -. A. x ph. (Contributed by NM, 21-May-2008.) Use its alias hbn1 1888 instead. (New usage is discouraged.)
|- ( -. A. x ph -> A. x -. A. x ph )
Theorem	hbn1 1888	Alias for ax-10 1887 to be used instead of it. (Contributed by NM, 24-Jan-1993.) (Proof shortened by Wolf Lammen, 18-Aug-2014.)
|- ( -. A. x ph -> A. x -. A. x ph )
Theorem	hbe1 1889	x is not free in E. x ph. (Contributed by NM, 24-Jan-1993.)
|- ( E. x ph -> A. x E. x ph )
Theorem	nfe1 1890	x is not free in E. x ph. (Contributed by Mario Carneiro, 11-Aug-2016.)
|- F/ x E. x ph
Theorem	modal-5 1891	The analog in our predicate calculus of axiom 5 of modal logic S5. (Contributed by NM, 5-Oct-2005.)
|- ( -. A. x -. ph -> A. x -. A. x -. ph )
1.5.2  Axiom scheme ax-11 (Quantifier Commutation)
Axiom	ax-11 1892	Axiom of Quantifier Commutation. This axiom says universal quantifiers can be swapped. Axiom scheme C6' in [Megill] p. 448 (p. 16 of the preprint). Also appears as Lemma 12 of [Monk2] p. 109 and Axiom C5-3 of [Monk2] p. 113. This axiom scheme is logically redundant (see ax11w 1876) but is used as an auxiliary axiom scheme to achieve metalogical completeness. (Contributed by NM, 12-Mar-1993.)
|- ( A. x A. y ph -> A. y A. x ph )
Theorem	alcoms 1893	Swap quantifiers in an antecedent. (Contributed by NM, 11-May-1993.)
|- ( A. x A. y ph -> ps )   =>    |- ( A. y A. x ph -> ps )
Theorem	hbal 1894	If x is not free in ph, it is not free in A. y ph. (Contributed by NM, 12-Mar-1993.)
|- ( ph -> A. x ph )   =>    |- ( A. y ph -> A. x A. y ph )
Theorem	alcom 1895	Theorem 19.5 of [Margaris] p. 89. (Contributed by NM, 30-Jun-1993.)
|- ( A. x A. y ph <-> A. y A. x ph )
Theorem	alrot3 1896	Theorem *11.21 in [WhiteheadRussell] p. 160. (Contributed by Andrew Salmon, 24-May-2011.)
|- ( A. x A. y A. z ph <-> A. y A. z A. x ph )
Theorem	alrot4 1897	Rotate four universal quantifiers twice. (Contributed by NM, 2-Feb-2005.) (Proof shortened by Fan Zheng, 6-Jun-2016.)
|- ( A. x A. y A. z A. w ph <-> A. z A. w A. x A. y ph )
Theorem	hbald 1898	Deduction form of bound-variable hypothesis builder hbal 1894. (Contributed by NM, 2-Jan-2002.)
|- ( ph -> A. y ph )   &    |- ( ph -> ( ps -> A. x ps ) )   =>    |- ( ph -> ( A. y ps -> A. x A. y ps ) )
Theorem	excom 1899	Theorem 19.11 of [Margaris] p. 89. (Contributed by NM, 5-Aug-1993.) Remove dependencies on ax-12 1905, ax-10 1887, ax-6 1794, ax-7 1839 and ax-5 1748. (Revised by Wolf Lammen, 8-Jan-2018.) (Proof shortened by Wolf Lammen, 22-Aug-2020.)
|- ( E. x E. y ph <-> E. y E. x ph )
Theorem	excomOLD 1900	Obsolete proof of excom 1899 as of 22-Aug-2020. (Contributed by NM, 5-Aug-1993.) Remove dependencies on ax-12 1905, ax-10 1887, ax-6 1794, ax-7 1839 and ax-5 1748. (Revised by Wolf Lammen, 8-Jan-2018.) (Proof modification is discouraged.) (New usage is discouraged.)
|- ( E. x E. y ph <-> E. y E. x ph )