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Theorem List for Metamath Proof Explorer - 1801-1900   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theorem19.9v 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 )
 
Theorem19.3v 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 )
 
Theoremspvw 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 )
 
Theorem19.39 1804 Theorem 19.39 of [Margaris] p. 90. (Contributed by NM, 12-Mar-1993.)
 |-  ( ( E. x ph 
 ->  E. x ps )  ->  E. x ( ph  ->  ps ) )
 
Theorem19.24 1805 Theorem 19.24 of [Margaris] p. 90. (Contributed by NM, 12-Mar-1993.)
 |-  ( ( A. x ph 
 ->  A. x ps )  ->  E. x ( ph  ->  ps ) )
 
Theorem19.34 1806 Theorem 19.34 of [Margaris] p. 90. (Contributed by NM, 12-Mar-1993.)
 |-  ( ( A. x ph 
 \/  E. x ps )  ->  E. x ( ph  \/  ps ) )
 
Theorem19.23v 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 ) )
 
Theorem19.23vv 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 )
 )
 
Theorem19.36v 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 ) )
 
Theorem19.36iv 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 )
 
Theorempm11.53v 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 )
 )
 
Theorem19.12vvv 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 )
 )
 
Theorem19.27v 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 ) )
 
Theorem19.28v 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 ) )
 
Theorem19.37v 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 ) )
 
Theorem19.37iv 1816* Inference associated with 19.37v 1815. (Contributed by NM, 5-Aug-1993.)
 |- 
 E. x ( ph  ->  ps )   =>    |-  ( ph  ->  E. x ps )
 
Theorem19.44v 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 ) )
 
Theorem19.45v 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 ) )
 
Theorem19.41v 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 ) )
 
Theorem19.41vv 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 )
 )
 
Theorem19.41vvv 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 ) )
 
Theorem19.41vvvv 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 ) )
 
Theorem19.42v 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 ) )
 
Theoremexdistr 1824* Distribution of existential quantifiers. (Contributed by NM, 9-Mar-1995.)
 |-  ( E. x E. y ( ph  /\  ps ) 
 <-> 
 E. x ( ph  /\ 
 E. y ps )
 )
 
Theorem19.42vv 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 )
 )
 
Theorem19.42vvv 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 ) )
 
Theoremexdistr2 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 ) )
 
Theorem3exdistr 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 ) ) )
 
Theorem4exdistr 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 )
 ) ) )
 
Theoremspimeh 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 )
 
Theoremspimw 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 )
 
Theoremspimvw 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 )
 
Theoremspnfw 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 )
 
Theoremspfalw 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 )
 
Theoremequs4v 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 ) )
 
Theoremequsalvw 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 )
 
Theoremcbvaliw 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 )
 
Theoremcbvalivw 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)
 
Axiomax-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
 ) )
 
Theoremequid 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
 
TheoremequidOLD 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
 
Theoremnfequid 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
 
Theoremequcomi 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 )
 
Theoremequcom 1844 Commutative law for equality. (Contributed by NM, 20-Aug-1993.)
 |-  ( x  =  y  <-> 
 y  =  x )
 
Theoremequcoms 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 )
 
Theoremequtr 1846 A transitive law for equality. (Contributed by NM, 23-Aug-1993.)
 |-  ( x  =  y 
 ->  ( y  =  z 
 ->  x  =  z
 ) )
 
Theoremequtrr 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
 ) )
 
Theoremequequ1 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 ) )
 
Theoremequequ2 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 ) )
 
Theoremstdpc6 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
 
Theoremstdpc7 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 )
 )
 
Theoremequtr2 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 )
 
Theoremequviniv 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 ) )
 
Theoremequvin 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 ) )
 
Theoremax13b 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 ) ) ) )
 
Theoremspfw 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 )
 
Theoremspw 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 )
 
Theoremcbvalw 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 )
 
Theoremcbvalvw 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 )
 
Theoremcbvexvw 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 )
 
Theoremalcomiw 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 )
 
Theoremhbn1fw 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 )
 
Theoremhbn1w 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 )
 
Theoremhba1w 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 )
 
Theoremhbe1w 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 )
 
Theoremhbalw 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 )
 
Theoremcbvaev 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
 
Syntaxwcel 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
 
Theoremwel 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)
 
Axiomax-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 ) )
 
Theoremelequ1 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)
 
Axiomax-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 ) )
 
Theoremelequ2 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.

 
Theoremax6dgen 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
 
Theoremax10w 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 )
 
Theoremax11w 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 )
 
Theoremax11dgen 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 )
 
Theoremax12wlem 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 )
 ) )
 
Theoremax12w 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 ) ) )
 
Theoremax12dgen 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 )
 ) )
 
Theoremax12wdemo 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 )
 ) ) )
 
Theoremax13w 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 )
 )
 
Theoremax13dgen1 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 )
 )
 
Theoremax13dgen2 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 )
 )
 
Theoremax13dgen3 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 )
 )
 
Theoremax13dgen4 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)
 
Axiomax-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 )
 
Theoremhbn1 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 )
 
Theoremhbe1 1889  x is not free in  E. x ph. (Contributed by NM, 24-Jan-1993.)
 |-  ( E. x ph  ->  A. x E. x ph )
 
Theoremnfe1 1890  x is not free in  E. x ph. (Contributed by Mario Carneiro, 11-Aug-2016.)
 |- 
 F/ x E. x ph
 
Theoremmodal-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)
 
Axiomax-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 )
 
Theoremalcoms 1893 Swap quantifiers in an antecedent. (Contributed by NM, 11-May-1993.)
 |-  ( A. x A. y ph  ->  ps )   =>    |-  ( A. y A. x ph  ->  ps )
 
Theoremhbal 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 )
 
Theoremalcom 1895 Theorem 19.5 of [Margaris] p. 89. (Contributed by NM, 30-Jun-1993.)
 |-  ( A. x A. y ph  <->  A. y A. x ph )
 
Theoremalrot3 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 )
 
Theoremalrot4 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 )
 
Theoremhbald 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 ) )
 
Theoremexcom 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 )
 
TheoremexcomOLD 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 )
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330 32901-33000 331 33001-33100 332 33101-33200 333 33201-33300 334 33301-33400 335 33401-33500 336 33501-33600 337 33601-33700 338 33701-33800 339 33801-33900 340 33901-34000 341 34001-34100 342 34101-34200 343 34201-34300 344 34301-34400 345 34401-34500 346 34501-34600 347 34601-34700 348 34701-34800 349 34801-34900 350 34901-35000 351 35001-35100 352 35101-35200 353 35201-35300 354 35301-35400 355 35401-35500 356 35501-35600 357 35601-35700 358 35701-35800 359 35801-35900 360 35901-36000 361 36001-36100 362 36101-36200 363 36201-36300 364 36301-36400 365 36401-36500 366 36501-36600 367 36601-36700 368 36701-36800 369 36801-36900 370 36901-37000 371 37001-37100 372 37101-37200 373 37201-37300 374 37301-37400 375 37401-37500 376 37501-37600 377 37601-37700 378 37701-37800 379 37801-37900 380 37901-38000 381 38001-38100 382 38101-38200 383 38201-38300 384 38301-38400 385 38401-38500 386 38501-38600 387 38601-38700 388 38701-38800 389 38801-38900 390 38901-39000 391 39001-39100 392 39101-39200 393 39201-39300 394 39301-39400 395 39401-39500 396 39501-39600 397 39601-39700 398 39701-39800 399 39801-39814
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