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	19.42vv 1801*	Version of 19.42 2000 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 1802*	Version of 19.42 2000 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 1803*	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 1804*	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 1805*	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 1806*	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 1807*	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 1808*	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 1809	Weak version of sp 1883. 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 1810	Version of sp 1883 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 1811*	Version of equs4 2061 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	cbvaliw 1812*	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 1813*	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 1814	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 1820). 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 1815	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
Theorem	nfequid 1816	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 1817	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 1818	Commutative law for equality. (Contributed by NM, 20-Aug-1993.)
|- ( x = y <-> y = x )
Theorem	equcoms 1819	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 1820	A transitive law for equality. (Contributed by NM, 23-Aug-1993.)
|- ( x = y -> ( y = z -> x = z ) )
Theorem	equtrr 1821	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 1822	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 1823	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 1824	One of the two equality axioms of standard predicate calculus, called reflexivity of equality. (The other one is stdpc7 1825.) 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 1825	One of the two equality axioms of standard predicate calculus, called substitutivity of equality. (The other one is stdpc6 1824.) 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 1826	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 1827*	A specialized version of equvini 2113 with a distinct variable restriction. (Contributed by Wolf Lammen, 8-Sep-2018.)
|- ( x = y -> E. z ( x = z /\ y = z ) )
Theorem	equvin 1828*	A variable introduction law for equality. Lemma 15 of [Monk2] p. 109. (Contributed by NM, 9-Jan-1993.) Remove dependencies on ax-10 1861, ax-13 2026. (Revised by Wolf Lammen, 10-Jun-2019.)
|- ( x = y <-> E. z ( x = z /\ z = y ) )
Theorem	ax13b 1829	Two equivalent ways of expressing ax-13 2026. See the comment for ax-13 2026. (Contributed by NM, 2-May-2017.) (Proof shortened by Wolf Lammen, 26-Feb-2018.)
|- ( ( -. x = y -> ( y = z -> A. x y = z ) ) <-> ( -. x = y -> ( -. x = z -> ( y = z -> A. x y = z ) ) ) )
Theorem	spfw 1830*	Weak version of sp 1883. 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 1831*	Weak version of the specialization scheme sp 1883. Lemma 9 of [KalishMontague] p. 87. While it appears that sp 1883 in its general form does not follow from Tarski's FOL axiom schemes, from this theorem we can prove any instance of sp 1883 having mutually distinct setvar variables and no wff metavariables (see ax12wdemo 1855 for an example of the procedure to eliminate the hypothesis). Other approximations of sp 1883 are spfw 1830 (minimal distinct variable requirements), spnfw 1809 (when x is not free in -. ph), spvw 1780 (when x does not appear in ph), sptruw 1651 (when ph is true), and spfalw 1810 (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 1832*	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 1833*	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 1834*	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 1835*	Weak version of alcom 1869. 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 1836*	Weak version of ax-10 1861 from which we can prove any ax-10 1861 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 1837*	Weak version of hbn1 1862. 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 1838*	Weak version of hba1 1924. See comments for ax10w 1849. 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 1839*	Weak version of hbe1 1863. See comments for ax10w 1849. 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 1840*	Weak version of hbal 1868. Uses only Tarski's FOL axiom schemes. Unlike hbal 1868, 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 1841*	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 1842	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 1843 of predicate calculus in terms of the wcel 1842 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 2388 for more information on the set theory usage of wcel 1842.)
wff A e. B
Theorem	wel 1843	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 1843 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 1842. This lets us avoid overloading the e. connective, thus preventing ambiguity that would complicate certain Metamath parsers. However, logically wel 1843 is considered to be a primitive syntax, even though here it is artificially "derived" from wcel 1842. 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 1844	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 1845	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 1846	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 1847	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 1652, ax-5 1725, ax6v 1772, ax-7 1814, ax-8 1844, and ax-9 1846, together with rule ax-gen 1639. See http://us.metamath.org/mpeuni/mmset.html#compare 1639. 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 1861, ax-11 1866, ax-12 1878, and ax-13 2026, 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 set metavariables, bundled or not, whose object-language instances are valid. (ax-12 1878 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 31907, but they can all be proved as theorems from the above.) Terminology: Two set (individual) metavariables are "bundled" in an axiom or theorem scheme when there is no distinct variable constraint ($d) imposed on them. (The term "bundled" is due to Raph Levien.) For example, the x and y in ax-6 1771 are bundled, but they are not in ax6v 1772. We also say that a scheme is bundled when it has at least one pair of bundled set metavariables. If distinct variable conditions are added to all set metavariable pairs in a bundled scheme, we call that the "principal" instance of the bundled scheme. For example, ax6v 1772 is the principal instance of ax-6 1771. 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 1771 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 1861, ax-11 1866, ax-12 1878, and ax-13 2026. "Translating" this to Metamath, it means that Tarski's axioms can prove any substitution instance of ax-10 1861, ax-11 1866, ax-12 1878, or ax-13 2026 in which (1) there are no wff metavariables and (2) all set metavariables are mutually distinct i.e. are not bundled. In effect this is mimicking the object language by pretending that each set metavariable is an object-language variable. (There may also be specific instances with wff metavariables and/or bundling that are directly provable from Tarski's axiom schemes, but it isn't guaranteed. Whether all of them are possible is part of the still open metalogical independence problem for our additional axiom schemes.) It can be useful to see how this can be done, both to show that our additional schemes are valid metatheorems of Tarski's system and to be able to translate object language instances of our proofs into proofs that would work with a system using only Tarski's original schemes. In addition, it may (or may not) provide insight into the conjectured metalogical independence of our additional schemes. The theorem schemes ax10w 1849, ax11w 1850, ax12w 1853, and ax13w 1856 are derived using only Tarski's axiom schemes, showing that Tarski's schemes can be used to derive all substitution instances of ax-10 1861, ax-11 1866, ax-12 1878, and ax-13 2026 meeting conditions (1) and (2). (The "w" suffix stands for "weak version".) Each hypothesis of ax10w 1849, ax11w 1850, and ax12w 1853 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 1855 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 1851, ax12dgen 1854, ax13dgen1 1857, ax13dgen2 1858, ax13dgen3 1859, and ax13dgen4 1860. (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 1861, ax-11 1866, ax-12 1878, and ax-13 2026 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 1771 in an older system, so it seems the main purpose of his later ax6v 1772 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 1771 as our official axiom, we show that the degenerate instance holds in ax6dgen 1848. The case of sp 1883 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 1831, again requiring substitution instances of ph that meet conditions (1) and (2) above. Note that our direct proof sp 1883 requires ax-12 1878, which is not part of Tarski's system.
Theorem	ax6dgen 1848	Tarski's system uses the weaker ax6v 1772 instead of the bundled ax-6 1771, so here we show that the degenerate case of ax-6 1771 can be derived. (Contributed by NM, 23-Apr-2017.)
|- -. A. x -. x = x
Theorem	ax10w 1849*	Weak version of ax-10 1861 from which we can prove any ax-10 1861 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 1850*	Weak version of ax-11 1866 from which we can prove any ax-11 1866 instance not involving wff variables or bundling. Uses only Tarski's FOL axiom schemes. Unlike ax-11 1866, 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 1851	Degenerate instance of ax-11 1866 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 1852*	Lemma for weak version of ax-12 1878. Uses only Tarski's FOL axiom schemes. In some cases, this lemma may lead to shorter proofs than ax12w 1853. (Contributed by NM, 10-Apr-2017.)
|- ( x = y -> ( ph <-> ps ) )   =>    |- ( x = y -> ( ph -> A. x ( x = y -> ph ) ) )
Theorem	ax12w 1853*	Weak version of ax-12 1878 from which we can prove any ax-12 1878 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 1855. (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 1854	Degenerate instance of ax-12 1878 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 1855*	Example of an application of ax12w 1853 that results in an instance of ax-12 1878 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 1833 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 1856*	Weak version (principal instance) of ax-13 2026. (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 1849, ax11w 1850, and ax12w 1853. (Contributed by NM, 10-Apr-2017.)
|- ( -. x = y -> ( y = z -> A. x y = z ) )
Theorem	ax13dgen1 1857	Degenerate instance of ax-13 2026 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 1858	Degenerate instance of ax-13 2026 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 1859	Degenerate instance of ax-13 2026 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 1860	Degenerate instance of ax-13 2026 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 1861, ax-11 1866, ax-12 1878, and ax-13 2026 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 set metavariables 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 1849, ax11w 1850, ax12w 1853, and ax13w 1856, which show that any object-language instance of these schemes (emulated by having no wff metavariables and requiring all set metavariables 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 1878 from all others has been shown, and independence of Tarski's ax-6 1771 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 1861	Axiom of Quantified Negation. Axiom C5-2 of [Monk2] p. 113. This axiom scheme is logically redundant (see ax10w 1849) but is used as an auxiliary axiom 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 1862 instead. (New usage is discouraged.)
|- ( -. A. x ph -> A. x -. A. x ph )
Theorem	hbn1 1862	Alias for ax-10 1861 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 1863	x is not free in E. x ph. (Contributed by NM, 24-Jan-1993.)
|- ( E. x ph -> A. x E. x ph )
Theorem	nfe1 1864	x is not free in E. x ph. (Contributed by Mario Carneiro, 11-Aug-2016.)
|- F/ x E. x ph
Theorem	modal-5 1865	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 1866	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 1850) but is used as an auxiliary axiom to achieve metalogical completeness. (Contributed by NM, 12-Mar-1993.)
|- ( A. x A. y ph -> A. y A. x ph )
Theorem	alcoms 1867	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 1868	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 1869	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 1870	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 1871	Rotate 4 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 1872	Deduction form of bound-variable hypothesis builder hbal 1868. (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 1873	Theorem 19.11 of [Margaris] p. 89. (Contributed by NM, 5-Aug-1993.) Remove dependencies on ax-12 1878, ax-10 1861, ax-6 1771, ax-7 1814 and ax-5 1725. (Revised by Wolf Lammen, 8-Jan-2018.)
|- ( E. x E. y ph <-> E. y E. x ph )
Theorem	excomim 1874	One direction of Theorem 19.11 of [Margaris] p. 89. (Contributed by NM, 5-Aug-1993.) (Revised by Mario Carneiro, 24-Sep-2016.) Remove dependencies on ax-12 1878, ax-10 1861, ax-6 1771, ax-7 1814 and ax-5 1725. (Revised by Wolf Lammen, 8-Jan-2018.)
|- ( E. x E. y ph -> E. y E. x ph )
Theorem	excom13 1875	Swap 1st and 3rd existential quantifiers. (Contributed by NM, 9-Mar-1995.)
|- ( E. x E. y E. z ph <-> E. z E. y E. x ph )
Theorem	exrot3 1876	Rotate existential quantifiers. (Contributed by NM, 17-Mar-1995.)
|- ( E. x E. y E. z ph <-> E. y E. z E. x ph )
Theorem	exrot4 1877	Rotate existential quantifiers twice. (Contributed by NM, 9-Mar-1995.)
|- ( E. x E. y E. z E. w ph <-> E. z E. w E. x E. y ph )
1.5.3  Axiom scheme ax-12 (Substitution)
Axiom	ax-12 1878	Axiom of Substitution. One of the 5 equality axioms of predicate calculus. The final consequent A. x ( x = y -> ph ) is a way of expressing " y substituted for x in wff ph " (cf. sb6 2197). It is based on Lemma 16 of [Tarski] p. 70 and Axiom C8 of [Monk2] p. 105, from which it can be proved by cases. The original version of this axiom was ax-c15 31913 and was replaced with this shorter ax-12 1878 in Jan. 2007. The old axiom is proved from this one as theorem axc15 2111. Conversely, this axiom is proved from ax-c15 31913 as theorem ax12 31927. Juha Arpiainen proved the metalogical independence of this axiom (in the form of the older axiom ax-c15 31913) from the others on 19-Jan-2006. See item 9a at http://us.metamath.org/award2003.html. See ax12v 1879 and ax12v2 2109 for other equivalents of this axiom that (unlike this axiom) have distinct variable restrictions. This axiom scheme is logically redundant (see ax12w 1853) but is used as an auxiliary axiom to achieve metalogical completeness. (Contributed by NM, 22-Jan-2007.)
|- ( x = y -> ( A. y ph -> A. x ( x = y -> ph ) ) )
Theorem	ax12v 1879*	This is a version of ax-12 1878 when the variables are distinct. Axiom (C8) of [Monk2] p. 105. See theorem ax12v2 2109 for the rederivation of ax-c15 31913 from this theorem. (Contributed by NM, 5-Aug-1993.) Removed dependencies on ax-10 1861 and ax-13 2026. (Revised by Jim Kingdon, 15-Dec-2017.) (Proof shortened by Wolf Lammen, 8-Dec-2019.)
|- ( x = y -> ( ph -> A. x ( x = y -> ph ) ) )
Theorem	ax12vOLD 1880*	Obsolete proof of ax12v 1879 as of 8-Dec-2019. (Contributed by Jim Kingdon, 15-Dec-2017.) (Proof modification is discouraged.) (New usage is discouraged.)
|- ( x = y -> ( ph -> A. x ( x = y -> ph ) ) )
Theorem	19.8a 1881	If a wff is true, it is true for at least one instance. Special case of Theorem 19.8 of [Margaris] p. 89. See 19.8v 1777 for a version requiring fewer axioms. (Contributed by NM, 9-Jan-1993.) Allow a shortening of sp 1883. (Revised by Wolf Lammen, 13-Jan-2018.) (Proof shortened by Wolf Lammen, 8-Dec-2019.)
|- ( ph -> E. x ph )
Theorem	19.8aOLD 1882	Obsolete proof of 19.8a 1881 as of 8-Dec-2019. (Contributed by NM, 9-Jan-1993.) (Revised by Wolf Lammen, 13-Jan-2018.) (Proof modification is discouraged.) (New usage is discouraged.)
|- ( ph -> E. x ph )
Theorem	sp 1883	Specialization. A universally quantified wff implies the wff without a quantifier Axiom scheme B5 of [Tarski] p. 67 (under his system S2, defined in the last paragraph on p. 77). Also appears as Axiom scheme C5' in [Megill] p. 448 (p. 16 of the preprint). For the axiom of specialization presented in many logic textbooks, see theorem stdpc4 2118. This theorem shows that our obsolete axiom ax-c5 31907 can be derived from the others. The proof uses ideas from the proof of Lemma 21 of [Monk2] p. 114. It appears that this scheme cannot be derived directly from Tarski's axioms without auxiliary axiom scheme ax-12 1878. It is thought the best we can do using only Tarski's axioms is spw 1831. (Contributed by NM, 21-May-2008.) (Proof shortened by Scott Fenton, 24-Jan-2011.) (Proof shortened by Wolf Lammen, 13-Jan-2018.)
|- ( A. x ph -> ph )
Theorem	axc4 1884	Show that the original axiom ax-c4 31908 can be derived from ax-4 1652 and others. See ax4 31918 for the rederivation of ax-4 1652 from ax-c4 31908. Part of the proof is based on the proof of Lemma 22 of [Monk2] p. 114. (Contributed by NM, 21-May-2008.) (Proof modification is discouraged.)
|- ( A. x ( A. x ph -> ps ) -> ( A. x ph -> A. x ps ) )
Theorem	axc7 1885	Show that the original axiom ax-c7 31909 can be derived from ax-10 1861 and others. See ax10 31919 for the rederivation of ax-10 1861 from ax-c7 31909. Normally, axc7 1885 should be used rather than ax-c7 31909, except by theorems specifically studying the latter's properties. (Contributed by NM, 21-May-2008.)
|- ( -. A. x -. A. x ph -> ph )
Theorem	axc7e 1886	Abbreviated version of axc7 1885. (Contributed by NM, 5-Aug-1993.)
|- ( E. x A. x ph -> ph )
Theorem	modal-b 1887	The analog in our predicate calculus of the Brouwer axiom (B) of modal logic S5. (Contributed by NM, 5-Oct-2005.)
|- ( ph -> A. x -. A. x -. ph )
Theorem	spi 1888	Inference rule reversing generalization. (Contributed by NM, 5-Aug-1993.)
|- A. x ph   =>    |- ph
Theorem	sps 1889	Generalization of antecedent. (Contributed by NM, 5-Jan-1993.)
|- ( ph -> ps )   =>    |- ( A. x ph -> ps )
Theorem	2sp 1890	A double specialization (see sp 1883). Another double specialization, closer to PM*11.1, is 2stdpc4 2119. (Contributed by BJ, 15-Sep-2018.)
|- ( A. x A. y ph -> ph )
Theorem	spsd 1891	Deduction generalizing antecedent. (Contributed by NM, 17-Aug-1994.)
|- ( ph -> ( ps -> ch ) )   =>    |- ( ph -> ( A. x ps -> ch ) )
Theorem	19.2g 1892	Theorem 19.2 of [Margaris] p. 89, generalized to use two setvar variables. Use 19.2 1775 when sufficient. (Contributed by Mel L. O'Cat, 31-Mar-2008.)
|- ( A. x ph -> E. y ph )
Theorem	19.21bi 1893	Inference form of 19.21 1933 and also deduction form of sp 1883. (Contributed by NM, 26-May-1993.)
|- ( ph -> A. x ps )   =>    |- ( ph -> ps )
Theorem	19.21bbi 1894	Inference removing double quantifier. Version of 19.21bi 1893 with two quanditiers. (Contributed by NM, 20-Apr-1994.)
|- ( ph -> A. x A. y ps )   =>    |- ( ph -> ps )
Theorem	19.23bi 1895	Inference form of Theorem 19.23 of [Margaris] p. 90, see 19.23 1938. (Contributed by NM, 12-Mar-1993.)
|- ( E. x ph -> ps )   =>    |- ( ph -> ps )
Theorem	nexr 1896	Inference form of 19.8a 1881. (Contributed by Jeff Hankins, 26-Jul-2009.)
|- -. E. x ph   =>    |- -. ph
Theorem	nfr 1897	Consequence of the definition of not-free. (Contributed by Mario Carneiro, 26-Sep-2016.)
|- ( F/ x ph -> ( ph -> A. x ph ) )
Theorem	nfri 1898	Consequence of the definition of not-free. (Contributed by Mario Carneiro, 11-Aug-2016.)
|- F/ x ph   =>    |- ( ph -> A. x ph )
Theorem	nfrd 1899	Consequence of the definition of not-free in a context. (Contributed by Mario Carneiro, 11-Aug-2016.)
|- ( ph -> F/ x ps )   =>    |- ( ph -> ( ps -> A. x ps ) )
Theorem	alimd 1900	Deduction form of Theorem 19.20 of [Margaris] p. 90, see alim 1653. (Contributed by Mario Carneiro, 24-Sep-2016.)
|- F/ x ph   &    |- ( ph -> ( ps -> ch ) )   =>    |- ( ph -> ( A. x ps -> A. x ch ) )