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Theorem List for Metamath Proof Explorer - 1801-1900   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremspeimfw 1801 Specialization, with additional weakening (compared to 19.2 1817) to allow bundling of  x and  y. Uses only Tarski's FOL axiom schemes. (Contributed by NM, 23-Apr-2017.) (Proof shortened by Wolf Lammen, 5-Dec-2017.)
 |-  ( x  =  y 
 ->  ( ph  ->  ps )
 )   =>    |-  ( -.  A. x  -.  x  =  y  ->  ( A. x ph  ->  E. x ps )
 )
 
TheoremspeimfwALT 1802 Alternate proof of speimfw 1801 (longer compressed proof, but fewer essential steps). (Contributed by NM, 23-Apr-2017.) (Proof shortened by Wolf Lammen, 5-Aug-2017.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  ( x  =  y 
 ->  ( ph  ->  ps )
 )   =>    |-  ( -.  A. x  -.  x  =  y  ->  ( A. x ph  ->  E. x ps )
 )
 
Theoremspimfw 1803 Specialization, with additional weakening (compared to sp 1957) to allow bundling of  x and  y. Uses only Tarski's FOL axiom schemes. (Contributed by NM, 23-Apr-2017.) (Proof shortened by Wolf Lammen, 7-Aug-2017.)
 |-  ( -.  ps  ->  A. x  -.  ps )   &    |-  ( x  =  y  ->  (
 ph  ->  ps ) )   =>    |-  ( -.  A. x  -.  x  =  y 
 ->  ( A. x ph  ->  ps ) )
 
Theoremax12i 1804 Inference that has ax-12 1950 (without  A. y) as its conclusion. Uses only Tarski's FOL axiom schemes. The hypotheses may be eliminable without using ax-12 1950 in special cases. Proof similar to Lemma 16 of [Tarski] p. 70. (Contributed by NM, 20-May-2008.)
 |-  ( x  =  y 
 ->  ( ph  <->  ps ) )   &    |-  ( ps  ->  A. x ps )   =>    |-  ( x  =  y  ->  (
 ph  ->  A. x ( x  =  y  ->  ph )
 ) )
 
1.4.6  Define proper substitution
 
Syntaxwsb 1805 Extend wff definition to include proper substitution (read "the wff that results when  y is properly substituted for  x in wff  ph"). (Contributed by NM, 24-Jan-2006.)
 wff  [ y  /  x ] ph
 
Definitiondf-sb 1806 Define proper substitution. Remark 9.1 in [Megill] p. 447 (p. 15 of the preprint). For our notation, we use  [ y  /  x ] ph to mean "the wff that results from the proper substitution of  y for  x in the wff  ph." That is,  y properly replaces  x. For example,  [ x  / 
y ] z  e.  y is the same as  z  e.  x, as shown in elsb4 2284. We can also use  [ y  /  x ] ph in place of the "free for" side condition used in traditional predicate calculus; see, for example, stdpc4 2202.

Our notation was introduced in Haskell B. Curry's Foundations of Mathematical Logic (1977), p. 316 and is frequently used in textbooks of lambda calculus and combinatory logic. This notation improves the common but ambiguous notation, " ph ( y ) is the wff that results when  y is properly substituted for  x in  ph ( x )." For example, if the original  ph ( x ) is  x  =  y, then  ph ( y ) is  y  =  y, from which we obtain that  ph ( x ) is  x  =  x. So what exactly does  ph ( x ) mean? Curry's notation solves this problem.

In most books, proper substitution has a somewhat complicated recursive definition with multiple cases based on the occurrences of free and bound variables in the wff. Instead, we use a single formula that is exactly equivalent and gives us a direct definition. We later prove that our definition has the properties we expect of proper substitution (see theorems sbequ 2225, sbcom2 2294 and sbid2v 2306).

Note that our definition is valid even when  x and  y are replaced with the same variable, as sbid 2101 shows. We achieve this by having  x free in the first conjunct and bound in the second. We can also achieve this by using a dummy variable, as the alternate definition dfsb7 2304 shows (which some logicians may prefer because it doesn't mix free and bound variables). Another version that mixes free and bound variables is dfsb3 2223. When  x and  y are distinct, we can express proper substitution with the simpler expressions of sb5 2279 and sb6 2278.

There are no restrictions on any of the variables, including what variables may occur in wff 
ph. (Contributed by NM, 10-May-1993.)

 |-  ( [ y  /  x ] ph  <->  ( ( x  =  y  ->  ph )  /\  E. x ( x  =  y  /\  ph )
 ) )
 
Theoremsbequ2 1807 An equality theorem for substitution. (Contributed by NM, 16-May-1993.) (Proof shortened by Wolf Lammen, 25-Feb-2018.)
 |-  ( x  =  y 
 ->  ( [ y  /  x ] ph  ->  ph )
 )
 
Theoremsb1 1808 One direction of a simplified definition of substitution. The converse requires either a dv condition (sb5 2279) or a non-freeness hypothesis (sb5f 2235). (Contributed by NM, 13-May-1993.)
 |-  ( [ y  /  x ] ph  ->  E. x ( x  =  y  /\  ph ) )
 
Theoremspsbe 1809 A specialization theorem. (Contributed by NM, 29-Jun-1993.) (Proof shortened by Wolf Lammen, 3-May-2018.)
 |-  ( [ y  /  x ] ph  ->  E. x ph )
 
Theoremsbequ8 1810 Elimination of equality from antecedent after substitution. (Contributed by NM, 5-Aug-1993.) Reduce dependencies on axioms. (Revised by Wolf Lammen, 28-Jul-2018.)
 |-  ( [ y  /  x ] ph  <->  [ y  /  x ] ( x  =  y  ->  ph ) )
 
Theoremsbimi 1811 Infer substitution into antecedent and consequent of an implication. (Contributed by NM, 25-Jun-1998.)
 |-  ( ph  ->  ps )   =>    |-  ( [ y  /  x ] ph  ->  [ y  /  x ] ps )
 
Theoremsbbii 1812 Infer substitution into both sides of a logical equivalence. (Contributed by NM, 14-May-1993.)
 |-  ( ph  <->  ps )   =>    |-  ( [ y  /  x ] ph  <->  [ y  /  x ] ps )
 
1.4.7  Axiom scheme ax-6 (Existence)
 
Axiomax-6 1813 Axiom of Existence. One of the equality and substitution axioms of predicate calculus with equality. This axiom tells us is that at least one thing exists. In this form (not requiring that  x and  y be distinct) it was used in an axiom system of Tarski (see Axiom B7' in footnote 1 of [KalishMontague] p. 81.) It is equivalent to axiom scheme C10' in [Megill] p. 448 (p. 16 of the preprint); the equivalence is established by axc10 2109 and ax6fromc10 32532. A more convenient form of this axiom is ax6e 2107, which has additional remarks.

Raph Levien proved the independence of this axiom from the other logical axioms on 12-Apr-2005. See item 16 at http://us.metamath.org/award2003.html.

ax-6 1813 can be proved from the weaker version ax6v 1814 requiring that the variables be distinct; see theorem ax6 2108.

ax-6 1813 can also be proved from the Axiom of Separation (in the form that we use that axiom, where free variables are not universally quantified). See theorem ax6vsep 4522.

Except by ax6v 1814, this axiom should not be referenced directly. Instead, use theorem ax6 2108. (Contributed by NM, 10-Jan-1993.) (New usage is discouraged.)

 |- 
 -.  A. x  -.  x  =  y
 
Theoremax6v 1814* Axiom B7 of [Tarski] p. 75, which requires that  x and  y be distinct. This trivial proof is intended merely to weaken axiom ax-6 1813 by adding a distinct variable restriction. From here on, ax-6 1813 should not be referenced directly by any other proof, so that theorem ax6 2108 will show that we can recover ax-6 1813 from this weaker version if it were an axiom (as it is in the case of Tarski).

Note: Introducing  x ,  y as a distinct variable group "out of the blue" with no apparent justification has puzzled some people, but it is perfectly sound. All we are doing is adding an additional redundant requirement, no different from adding a redundant logical hypothesis, that results in a weakening of the theorem. This means that any future theorem that references ax6v 1814 must have a $d specified for the two variables that get substituted for  x and  y. The $d does not propagate "backwards" i.e. it does not impose a requirement on ax-6 1813.

When possible, use of this theorem rather than ax6 2108 is preferred since its derivation is much shorter and requires fewer axioms. (Contributed by NM, 7-Aug-2015.)

 |- 
 -.  A. x  -.  x  =  y
 
Theoremax6ev 1815* At least one individual exists. Weaker version of ax6e 2107. When possible, use of this theorem rather than ax6e 2107 is preferred since its derivation is much shorter and requires fewer axioms. (Contributed by NM, 3-Aug-2017.)
 |- 
 E. x  x  =  y
 
Theoremexiftru 1816 Rule of existential generalization, similar to universal generalization ax-gen 1677, but valid only if an individual exists. Its proof requires ax-6 1813 but the equality predicate does not occur in its statement. Some fundamental theorems of predicate logic can be proven from ax-gen 1677, ax-4 1690 and this theorem alone, not requiring ax-7 1859 or excessive distinct variable conditions. (Contributed by Wolf Lammen, 12-Nov-2017.) (Proof shortened by Wolf Lammen, 9-Dec-2017.)
 |-  ph   =>    |- 
 E. x ph
 
Theorem19.2 1817 Theorem 19.2 of [Margaris] p. 89. This corresponds to the axiom (D) of modal logic. Note: This proof is very different from Margaris' because we only have Tarski's FOL axiom schemes available at this point. See the later 19.2g 1966 for a more conventional proof of a more general result, which uses additional axioms. (Contributed by NM, 2-Aug-2017.) Remove dependency on ax-7 1859. (Revised by Wolf Lammen, 4-Dec-2017.)
 |-  ( A. x ph  ->  E. x ph )
 
Theorem19.8w 1818 Weak version of 19.8a 1955. Uses only Tarski's FOL axiom schemes. (Contributed by NM, 1-Aug-2017.) (Proof shortened by Wolf Lammen, 4-Dec-2017.)
 |-  ( ph  ->  A. x ph )   =>    |-  ( ph  ->  E. x ph )
 
Theorem19.8v 1819* Version of 19.8a 1955 with a dv condition, requiring fewer axioms. (Contributed by BJ, 12-Mar-2020.)
 |-  ( ph  ->  E. x ph )
 
Theorem19.9v 1820* Version of 19.9 1990 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 1821. (Contributed by NM, 28-May-1995.) Remove dependency on ax-7 1859. (Revised by Wolf Lammen, 4-Dec-2017.)
 |-  ( E. x ph  <->  ph )
 
Theorem19.3v 1821* Version of 19.3 1986 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 1820. (Contributed by Anthony Hart, 13-Sep-2011.) Remove dependency on ax-7 1859. (Revised by Wolf Lammen, 4-Dec-2017.)
 |-  ( A. x ph  <->  ph )
 
Theoremspvw 1822* Version of sp 1957 when  x does not occur in  ph. Converse of ax-5 1766. 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 1823 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 1824 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 1825 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 1826* Version of 19.23 2013 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 1827* Theorem 19.23v 1826 extended to two variables. (Contributed by NM, 10-Aug-2004.)
 |-  ( A. x A. y ( ph  ->  ps )  <->  ( E. x E. y ph  ->  ps )
 )
 
Theorem19.36v 1828* Version of 19.36 2063 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 1829* Inference associated with 19.36v 1828. Version of 19.36i 2064 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 1830* Version of pm11.53 2090 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 1831* Version of 19.12vv 2091 with a dv condition, requiring fewer axioms. See also 19.12 2052. (Contributed by BJ, 18-Mar-2020.)
 |-  ( E. x A. y ( ph  ->  ps )  <->  A. y E. x ( ph  ->  ps )
 )
 
Theorem19.27v 1832* Version of 19.27 2026 with a dv condition, requiring fewer axioms. (Contributed by NM, 3-Jun-2004.)
 |-  ( A. x (
 ph  /\  ps )  <->  (
 A. x ph  /\  ps ) )
 
Theorem19.28v 1833* Version of 19.28 2027 with a dv condition, requiring fewer axioms. (Contributed by NM, 25-Mar-2004.)
 |-  ( A. x (
 ph  /\  ps )  <->  (
 ph  /\  A. x ps ) )
 
Theorem19.37v 1834* Version of 19.37 2065 with a dv condition, requiring fewer axioms. (Contributed by NM, 21-Jun-1993.)
 |-  ( E. x (
 ph  ->  ps )  <->  ( ph  ->  E. x ps ) )
 
Theorem19.37iv 1835* Inference associated with 19.37v 1834. (Contributed by NM, 5-Aug-1993.)
 |- 
 E. x ( ph  ->  ps )   =>    |-  ( ph  ->  E. x ps )
 
Theorem19.44v 1836* Version of 19.44 2068 with a dv condition, requiring fewer axioms. (Contributed by NM, 12-Mar-1993.)
 |-  ( E. x (
 ph  \/  ps )  <->  ( E. x ph  \/  ps ) )
 
Theorem19.45v 1837* Version of 19.45 2069 with a dv condition, requiring fewer axioms. (Contributed by NM, 12-Mar-1993.)
 |-  ( E. x (
 ph  \/  ps )  <->  (
 ph  \/  E. x ps ) )
 
Theorem19.41v 1838* Version of 19.41 2070 with a dv condition, requiring fewer axioms. (Contributed by NM, 21-Jun-1993.)
 |-  ( E. x (
 ph  /\  ps )  <->  ( E. x ph  /\  ps ) )
 
Theorem19.41vv 1839* Version of 19.41 2070 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 1840* Version of 19.41 2070 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 1841* Version of 19.41 2070 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 1842* Version of 19.42 2071 with a dv condition requiring fewer axioms. (Contributed by NM, 21-Jun-1993.)
 |-  ( E. x (
 ph  /\  ps )  <->  (
 ph  /\  E. x ps ) )
 
Theoremexdistr 1843* Distribution of existential quantifiers. (Contributed by NM, 9-Mar-1995.)
 |-  ( E. x E. y ( ph  /\  ps ) 
 <-> 
 E. x ( ph  /\ 
 E. y ps )
 )
 
Theorem19.42vv 1844* Version of 19.42 2071 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 1845* Version of 19.42 2071 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 1846* 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 1847* 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 1848* 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 1849* 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 1850* 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 1851* 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 1852 Weak version of sp 1957. 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 1853 Version of sp 1957 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 1854* Version of equs4 2140 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 1855* Version of equsal 2141 with two dv conditions, which requires fewer axioms. (Contributed by BJ, 31-May-2019.)
 |-  ( x  =  y 
 ->  ( ph  <->  ps ) )   =>    |-  ( A. x ( x  =  y  -> 
 ph )  <->  ps )
 
Theoremequsexvw 1856* Version of equsexv 2085 with a dv condition, which requires fewer axioms. See also equsex 2143. (Contributed by BJ, 31-May-2019.)
 |-  ( x  =  y 
 ->  ( ph  <->  ps ) )   =>    |-  ( E. x ( x  =  y  /\  ph )  <->  ps )
 
Theoremcbvaliw 1857* 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 1858* 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 1859 Axiom of Equality. One of the equality and substitution axioms of predicate calculus with equality. It states that equality is right-Euclidean (this is similar, but not identical, to being transitive, which is proved as equtr 1873). 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 1557 by Robert Recorde. He chose a pair of parallel lines of the same length because "noe .2. thynges, can be moare equalle."

We prove in ax7 1868 that this axiom can be recovered from its weakened version ax7v 1860 where  x and  y are assumed to be disjoint variables. In particular, the only theorem referencing ax-7 1859 should be ax7v 1860. See the comment of ax7v 1860 for more details on these matters. (Contributed by NM, 10-Jan-1993.) (Revised by BJ, 7-Dec-2020.) Use ax7 1868 instead. (New usage is discouraged.)

 |-  ( x  =  y 
 ->  ( x  =  z 
 ->  y  =  z
 ) )
 
Theoremax7v 1860* Weakened version of ax-7 1859, with a dv condition on  x ,  y. This should be the only proof referencing ax-7 1859, and it should be referenced only by its two weakened versions ax7v1 1861 and ax7v2 1862, from which ax-7 1859 is then rederived as ax7 1868, which shows that either ax7v 1860 or the conjunction of ax7v1 1861 and ax7v2 1862 is sufficient.

In ax7v 1860, it is still allowed to substitute the same variable for  x and  z, or the same variable for  y and  z. Therefore, ax7v 1860 "bundles" (a term coined by Raph Levien) its "principal instance"  ( x  =  y  ->  ( x  =  z  ->  y  =  z ) ) with 
x ,  y ,  z distinct, and its "degenerate instances"  ( x  =  y  ->  ( x  =  x  ->  y  =  x ) ) and  ( x  =  y  ->  ( x  =  y  ->  y  =  y ) ) with 
x ,  y distinct. These degenerate instances are for instance used in the proofs of equcomiv 1866 and equid 1863 respectively. (Contributed by BJ, 7-Dec-2020.) Use ax7 1868 instead. (New usage is discouraged.)

 |-  ( x  =  y 
 ->  ( x  =  z 
 ->  y  =  z
 ) )
 
Theoremax7v1 1861* First of two weakened versions of ax7v 1860, with an extra dv condition on  x ,  z, see comments there. (Contributed by BJ, 7-Dec-2020.)
 |-  ( x  =  y 
 ->  ( x  =  z 
 ->  y  =  z
 ) )
 
Theoremax7v2 1862* Second of two weakened versions of ax7v 1860, with an extra dv condition on  y ,  z, see comments there. (Contributed by BJ, 7-Dec-2020.)
 |-  ( x  =  y 
 ->  ( x  =  z 
 ->  y  =  z
 ) )
 
Theoremequid 1863 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 1864 Obsolete proof of equid 1863 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 1865 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
 
Theoremequcomiv 1866* Weaker form of equcomi 1869 with a dv condition on  x ,  y. This is an intermediate step and equcomi 1869 is fully recovered later. (Contributed by BJ, 7-Dec-2020.)
 |-  ( x  =  y 
 ->  y  =  x )
 
Theoremax6evr 1867* A commuted form of ax6ev 1815. (Contributed by BJ, 7-Dec-2020.)
 |- 
 E. x  y  =  x
 
Theoremax7 1868 Proof of ax-7 1859 from ax7v1 1861 and ax7v2 1862, proving sufficiency of the conjunction of the latter two weakened versions of ax7v 1860, which is itself a weakened version of ax-7 1859.

Note that the weakened version of ax-7 1859 obtained by adding a dv condition on  x ,  z (resp. on  y ,  z) does not permit, together with the other axioms, to prove reflexivity (resp. symmetry). (Contributed by BJ, 7-Dec-2020.)

 |-  ( x  =  y 
 ->  ( x  =  z 
 ->  y  =  z
 ) )
 
Theoremequcomi 1869 Commutative law for equality. Equality is a symmetric relation. 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 1870 Commutative law for equality. Equality is a symmetric relation. (Contributed by NM, 20-Aug-1993.)
 |-  ( x  =  y  <-> 
 y  =  x )
 
Theoremequcomd 1871 Deduction form of equcom 1870, symmetry of equality. For the versions for classes, see eqcom 2478 and eqcomd 2477. (Contributed by BJ, 6-Oct-2019.)
 |-  ( ph  ->  x  =  y )   =>    |-  ( ph  ->  y  =  x )
 
Theoremequcoms 1872 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 1873 A transitive law for equality. (Contributed by NM, 23-Aug-1993.)
 |-  ( x  =  y 
 ->  ( y  =  z 
 ->  x  =  z
 ) )
 
Theoremequtrr 1874 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 1875 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 1876 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 ) )
 
Theoremequtr2 1877 Equality is left-Euclidean. (Contributed by NM, 12-Aug-1993.) (Proof shortened by Andrew Salmon, 25-May-2011.)
 |-  ( ( x  =  z  /\  y  =  z )  ->  x  =  y )
 
Theoremstdpc6 1878 One of the two equality axioms of standard predicate calculus, called reflexivity of equality. (The other one is stdpc7 1879.) 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 1879 One of the two equality axioms of standard predicate calculus, called substitutivity of equality. (The other one is stdpc6 1878.) 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 )
 )
 
Theoremequviniv 1880* A specialized version of equvini 2195 with a distinct variable restriction. (Contributed by Wolf Lammen, 8-Sep-2018.)
 |-  ( x  =  y 
 ->  E. z ( x  =  z  /\  y  =  z ) )
 
Theoremequvin 1881* A variable introduction law for equality. Lemma 15 of [Monk2] p. 109. (Contributed by NM, 9-Jan-1993.) Remove dependencies on ax-10 1932, ax-13 2104. (Revised by Wolf Lammen, 10-Jun-2019.)
 |-  ( x  =  y  <->  E. z ( x  =  z  /\  z  =  y ) )
 
Theoremax13b 1882 An equivalence used to show two ways of expressing ax-13 2104. See the comment for ax-13 2104. (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 1883* Weak version of sp 1957. 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 1884* Weak version of the specialization scheme sp 1957. Lemma 9 of [KalishMontague] p. 87. While it appears that sp 1957 in its general form does not follow from Tarski's FOL axiom schemes, from this theorem we can prove any instance of sp 1957 having mutually distinct setvar variables and no wff metavariables (see ax12wdemo 1926 for an example of the procedure to eliminate the hypothesis). Other approximations of sp 1957 are spfw 1883 (minimal distinct variable requirements), spnfw 1852 (when  x is not free in  -.  ph), spvw 1822 (when  x does not appear in  ph), sptruw 1689 (when  ph is true), and spfalw 1853 (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 1885* 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 1886* 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 1887* 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 1888* Weak version of alcom 1940. 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 1889* Weak version of ax-10 1932 from which we can prove any ax-10 1932 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 1890* Weak version of hbn1 1933. 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 1891* Weak version of hba1 1998. See comments for ax10w 1920. 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 1892* Weak version of hbe1 1934. See comments for ax10w 1920. 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 1893* Weak version of hbal 1939. Uses only Tarski's FOL axiom schemes. Unlike hbal 1939, 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 )
 
Theoremspaev 1894* A special instance of sp 1957 applied to an equality with a dv condition. Unlike the more general sp 1957, we can prove this without ax-12 1950. This is an instance of aeveq 1898.

The antecedent  A. x x  =  y with distinct  x and  y is a characteristic of a degenerate universe, in which just one object exists. Actually more than one object may still exist, but if so, we give up on equality as a discriminating term.

Separating this degenerate case from a richer universe, where inequality is possible, is a common proof idea. The name of this theorem follows a convention, where the condition  A. x x  =  y is denoted by 'aev', a shorthand for 'all equal, with a distinct variable condition'. (Contributed by Wolf Lammen, 14-Mar-2021.)

 |-  ( A. x  x  =  y  ->  x  =  y )
 
Theoremcbvaev 1895* Change bound variable in an equality with a dv condition. This is an instance of aev 1899. (Contributed by NM, 22-Jul-2015.) (Revised by BJ, 18-Jun-2019.)
 |-  ( A. x  x  =  w  ->  A. y  y  =  w )
 
Theoremaevlem0 1896* Lemma for aevlem 1897. This is an instance of aev 1899. (Contributed by NM, 8-Jul-2016.) (Proof shortened by Wolf Lammen, 17-Feb-2018.) Remove dependency on ax-12 1950. (Revised by Wolf Lammen, 14-Mar-2021.) (Revised by BJ, 29-Mar-2021.)
 |-  ( A. x  x  =  z  ->  A. y  y  =  x )
 
Theoremaevlem 1897* Lemma for aev 1899 and axc16g 2042. Change free and bound variables. This is an instance of aev 1899. (Contributed by NM, 22-Jul-2015.) (Proof shortened by Wolf Lammen, 17-Feb-2018.) Remove dependency on ax-13 2104, along an idea of BJ. (Revised by Wolf Lammen, 30-Nov-2019.) (Revised by BJ, 29-Mar-2021.)
 |-  ( A. z  z  =  w  ->  A. y  y  =  x )
 
Theoremaeveq 1898*  A. x x  =  y with a dv restriction (One-object-universe) disallows inequality. (Contributed by Wolf Lammen, 19-Mar-2021.)
 |-  ( A. x  x  =  y  ->  w  =  v )
 
Theoremaev 1899* A "distinctor elimination" lemma with no restrictions on variables in the consequent. (Contributed by NM, 8-Nov-2006.) Remove dependency on ax-11 1937. (Revised by Wolf Lammen, 7-Sep-2018.) Remove dependency on ax-13 2104, inspired by an idea of BJ. (Revised by Wolf Lammen, 30-Nov-2019.) Remove dependency on ax-12 1950. (Revised by Wolf Lammen, 19-Mar-2021.)
 |-  ( A. x  x  =  y  ->  A. z  w  =  v )
 
Theoremhbaevg 1900* Generalization of hbaev 1901, proved at no extra cost. (Contributed by Wolf Lammen, 22-Mar-2021.) (Revised by BJ, 29-Mar-2021.)
 |-  ( A. x  x  =  y  ->  A. z A. u  u  =  t )
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