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Theorem List for Metamath Proof Explorer - 2001-2100   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremax10o 2001 Show that ax-10o 2189 can be derived from ax-10 2190 in the form of ax10 1991. Normally, ax10o 2001 should be used rather than ax-10o 2189, except by theorems specifically studying the latter's properties. (Contributed by NM, 16-May-2008.) (Proof modification is discouraged.)
 |-  ( A. x  x  =  y  ->  ( A. x ph  ->  A. y ph ) )
 
Theoremaecom 2002 Commutation law for identical variable specifiers. The antecedent and consequent are true when  x and  y are substituted with the same variable. Lemma L12 in [Megill] p. 445 (p. 12 of the preprint). (Contributed by NM, 5-Aug-1993.)
 |-  ( A. x  x  =  y  ->  A. y  y  =  x )
 
Theoremaecoms 2003 A commutation rule for identical variable specifiers. (Contributed by NM, 5-Aug-1993.)
 |-  ( A. x  x  =  y  ->  ph )   =>    |-  ( A. y  y  =  x  ->  ph )
 
Theoremnaecoms 2004 A commutation rule for distinct variable specifiers. (Contributed by NM, 2-Jan-2002.)
 |-  ( -.  A. x  x  =  y  ->  ph )   =>    |-  ( -.  A. y  y  =  x  ->  ph )
 
Theoremhbae 2005 All variables are effectively bound in an identical variable specifier. (Contributed by NM, 5-Aug-1993.)
 |-  ( A. x  x  =  y  ->  A. z A. x  x  =  y )
 
Theoremnfae 2006 All variables are effectively bound in an identical variable specifier. (Contributed by Mario Carneiro, 11-Aug-2016.)
 |- 
 F/ z A. x  x  =  y
 
Theoremhbnae 2007 All variables are effectively bound in a distinct variable specifier. Lemma L19 in [Megill] p. 446 (p. 14 of the preprint). (Contributed by NM, 5-Aug-1993.)
 |-  ( -.  A. x  x  =  y  ->  A. z  -.  A. x  x  =  y )
 
Theoremnfnae 2008 All variables are effectively bound in a distinct variable specifier. (Contributed by Mario Carneiro, 11-Aug-2016.)
 |- 
 F/ z  -.  A. x  x  =  y
 
Theoremhbnaes 2009 Rule that applies hbnae 2007 to antecedent. (Contributed by NM, 5-Aug-1993.)
 |-  ( A. z  -.  A. x  x  =  y 
 ->  ph )   =>    |-  ( -.  A. x  x  =  y  ->  ph )
 
Theoremaevlem1 2010* Lemma for aev 2011 and a16g 2012. Change free and bound variables. (Contributed by NM, 22-Jul-2015.) (Proof shortened by Wolf Lammen, 17-Feb-2018.)
 |-  ( A. z  z  =  w  ->  A. y  y  =  x )
 
Theoremaev 2011* A "distinctor elimination" lemma with no restrictions on variables in the consequent. (Contributed by NM, 8-Nov-2006.)
 |-  ( A. x  x  =  y  ->  A. z  w  =  v )
 
Theorema16g 2012* Generalization of ax16 2094. (Contributed by NM, 25-Jul-2015.) (Proof shortened by Wolf Lammen, 18-Feb-2018.)
 |-  ( A. x  x  =  y  ->  ( ph  ->  A. z ph )
 )
 
Theorema16gOLD 2013* Obsolete proof of a16g 2012 as of 18-Feb-2018. (Contributed by NM, 25-Jul-2015.) (New usage is discouraged.) (Proof modification is discouraged.)
 |-  ( A. x  x  =  y  ->  ( ph  ->  A. z ph )
 )
 
Theoremnfeqf 2014 A variable is effectively not free in an equality if it is not either of the involved variables.  F/ version of ax-12o 2192. (Contributed by Mario Carneiro, 6-Oct-2016.)
 |-  ( ( -.  A. z  z  =  x  /\  -.  A. z  z  =  y )  ->  F/ z  x  =  y )
 
Theoremdvelimh 2015 Version of dvelim 2066 without any variable restrictions. (Contributed by NM, 1-Oct-2002.) (Proof shortened by Wolf Lammen, 4-Mar-2018.)
 |-  ( ph  ->  A. x ph )   &    |-  ( ps  ->  A. z ps )   &    |-  (
 z  =  y  ->  ( ph  <->  ps ) )   =>    |-  ( -.  A. x  x  =  y  ->  ( ps  ->  A. x ps ) )
 
TheoremdvelimhOLD 2016 Obslete proof of dvelimh 2015 as of 4-Mar-2018. (Contributed by NM, 1-Oct-2002.) (New usage is discouraged.) (Proof modification is discouraged.)
 |-  ( ph  ->  A. x ph )   &    |-  ( ps  ->  A. z ps )   &    |-  (
 z  =  y  ->  ( ph  <->  ps ) )   =>    |-  ( -.  A. x  x  =  y  ->  ( ps  ->  A. x ps ) )
 
Theoremdvelimv 2017* Similar to dvelim 2066 with first hypothesis replaced by distinct variable condition. (Contributed by NM, 25-Jul-2015.) (Proof shortened by Wolf Lammen, 25-Feb-2018.)
 |-  ( z  =  y 
 ->  ( ph  <->  ps ) )   =>    |-  ( -.  A. x  x  =  y  ->  ( ps  ->  A. x ps ) )
 
Theoremdveeq1OLD 2018* Obsolete proof of dveeq1 1987 as of 25-Feb-2018. (Contributed by NM, 2-Jan-2002.) (New usage is discouraged.) (Proof modification is discouraged.)
 |-  ( -.  A. x  x  =  y  ->  ( y  =  z  ->  A. x  y  =  z ) )
 
Theoremdveeq2 2019* Quantifier introduction when one pair of variables is distinct. (Contributed by NM, 2-Jan-2002.) (Revised by NM, 20-Jul-2015.)
 |-  ( -.  A. x  x  =  y  ->  ( z  =  y  ->  A. x  z  =  y ) )
 
Theoremdral2 2020 Formula-building lemma for use with the Distinctor Reduction Theorem. Part of Theorem 9.4 of [Megill] p. 448 (p. 16 of preprint). (Contributed by NM, 27-Feb-2005.) (Revised by Wolf Lammen, 4-Mar-2018.)
 |-  ( A. x  x  =  y  ->  ( ph 
 <->  ps ) )   =>    |-  ( A. x  x  =  y  ->  (
 A. z ph  <->  A. z ps )
 )
 
Theoremdral2OLD 2021 Formula-building lemma for use with the Distinctor Reduction Theorem. Part of Theorem 9.4 of [Megill] p. 448 (p. 16 of preprint). (Contributed by NM, 27-Feb-2005.) (New usage is discouraged.) (Proof modification is discouraged.)
 |-  ( A. x  x  =  y  ->  ( ph 
 <->  ps ) )   =>    |-  ( A. x  x  =  y  ->  (
 A. z ph  <->  A. z ps )
 )
 
Theoremdral1 2022 Formula-building lemma for use with the Distinctor Reduction Theorem. Part of Theorem 9.4 of [Megill] p. 448 (p. 16 of preprint). (Contributed by NM, 24-Nov-1994.) (Proof shortened by Wolf Lammen, 4-Mar-2018.)
 |-  ( A. x  x  =  y  ->  ( ph 
 <->  ps ) )   =>    |-  ( A. x  x  =  y  ->  (
 A. x ph  <->  A. y ps )
 )
 
Theoremdral1OLD 2023 Obsolete proof of dral1 2022 as of 4-Mar-2018. (Contributed by NM, 24-Nov-1994.) (New usage is discouraged.) (Proof modification is discouraged.)
 |-  ( A. x  x  =  y  ->  ( ph 
 <->  ps ) )   =>    |-  ( A. x  x  =  y  ->  (
 A. x ph  <->  A. y ps )
 )
 
Theoremdrex1 2024 Formula-building lemma for use with the Distinctor Reduction Theorem. Part of Theorem 9.4 of [Megill] p. 448 (p. 16 of preprint). (Contributed by NM, 27-Feb-2005.)
 |-  ( A. x  x  =  y  ->  ( ph 
 <->  ps ) )   =>    |-  ( A. x  x  =  y  ->  ( E. x ph  <->  E. y ps )
 )
 
Theoremdrex2 2025 Formula-building lemma for use with the Distinctor Reduction Theorem. Part of Theorem 9.4 of [Megill] p. 448 (p. 16 of preprint). (Contributed by NM, 27-Feb-2005.)
 |-  ( A. x  x  =  y  ->  ( ph 
 <->  ps ) )   =>    |-  ( A. x  x  =  y  ->  ( E. z ph  <->  E. z ps )
 )
 
Theoremdrnf1 2026 Formula-building lemma for use with the Distinctor Reduction Theorem. (Contributed by Mario Carneiro, 4-Oct-2016.)
 |-  ( A. x  x  =  y  ->  ( ph 
 <->  ps ) )   =>    |-  ( A. x  x  =  y  ->  ( F/ x ph  <->  F/ y ps )
 )
 
Theoremdrnf2 2027 Formula-building lemma for use with the Distinctor Reduction Theorem. (Contributed by Mario Carneiro, 4-Oct-2016.)
 |-  ( A. x  x  =  y  ->  ( ph 
 <->  ps ) )   =>    |-  ( A. x  x  =  y  ->  ( F/ z ph  <->  F/ z ps )
 )
 
Theoremexdistrf 2028 Distribution of existential quantifiers, with a bound-variable hypothesis saying that  y is not free in  ph, but  x can be free in  ph (and there is no distinct variable condition on  x and  y). (Contributed by Mario Carneiro, 20-Mar-2013.)
 |-  ( -.  A. x  x  =  y  ->  F/ y ph )   =>    |-  ( E. x E. y ( ph  /\  ps )  ->  E. x ( ph  /\ 
 E. y ps )
 )
 
Theoremnfald2 2029 Variation on nfald 1867 which adds the hypothesis that  x and  y are distinct in the inner subproof. (Contributed by Mario Carneiro, 8-Oct-2016.)
 |- 
 F/ y ph   &    |-  ( ( ph  /\ 
 -.  A. x  x  =  y )  ->  F/ x ps )   =>    |-  ( ph  ->  F/ x A. y ps )
 
Theoremnfexd2 2030 Variation on nfexd 1869 which adds the hypothesis that  x and  y are distinct in the inner subproof. (Contributed by Mario Carneiro, 8-Oct-2016.)
 |- 
 F/ y ph   &    |-  ( ( ph  /\ 
 -.  A. x  x  =  y )  ->  F/ x ps )   =>    |-  ( ph  ->  F/ x E. y ps )
 
Theoremcbv1h 2031 Rule used to change bound variables, using implicit substitution. (Contributed by NM, 5-Aug-1993.)
 |-  ( ph  ->  ( ps  ->  A. y ps )
 )   &    |-  ( ph  ->  ( ch  ->  A. x ch )
 )   &    |-  ( ph  ->  ( x  =  y  ->  ( ps  ->  ch )
 ) )   =>    |-  ( A. x A. y ph  ->  ( A. x ps  ->  A. y ch ) )
 
Theoremcbv1 2032 Rule used to change bound variables, using implicit substitution. (Contributed by NM, 5-Aug-1993.) (Revised by Mario Carneiro, 3-Oct-2016.)
 |-  ( ph  ->  F/ y ps )   &    |-  ( ph  ->  F/ x ch )   &    |-  ( ph  ->  ( x  =  y  ->  ( ps  ->  ch ) ) )   =>    |-  ( A. x A. y ph  ->  ( A. x ps  ->  A. y ch )
 )
 
Theoremcbv2h 2033 Rule used to change bound variables, using implicit substitution. (Contributed by NM, 5-Aug-1993.)
 |-  ( ph  ->  ( ps  ->  A. y ps )
 )   &    |-  ( ph  ->  ( ch  ->  A. x ch )
 )   &    |-  ( ph  ->  ( x  =  y  ->  ( ps  <->  ch ) ) )   =>    |-  ( A. x A. y ph  ->  ( A. x ps 
 <-> 
 A. y ch )
 )
 
Theoremcbv2 2034 Rule used to change bound variables, using implicit substitution. (Contributed by NM, 5-Aug-1993.) (Revised by Mario Carneiro, 3-Oct-2016.)
 |-  ( ph  ->  F/ y ps )   &    |-  ( ph  ->  F/ x ch )   &    |-  ( ph  ->  ( x  =  y  ->  ( ps  <->  ch ) ) )   =>    |-  ( A. x A. y ph  ->  ( A. x ps  <->  A. y ch )
 )
 
Theoremcbv3 2035 Rule used to change bound variables, using implicit substitution, that does not use ax-12o 2192. (Contributed by NM, 5-Aug-1993.)
 |- 
 F/ y ph   &    |-  F/ x ps   &    |-  ( x  =  y  ->  (
 ph  ->  ps ) )   =>    |-  ( A. x ph 
 ->  A. y ps )
 
Theoremcbv3h 2036 Rule used to change bound variables, using implicit substitution. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Andrew Salmon, 25-May-2011.) (Proof modification is discouraged.)
 |-  ( ph  ->  A. y ph )   &    |-  ( ps  ->  A. x ps )   &    |-  ( x  =  y  ->  (
 ph  ->  ps ) )   =>    |-  ( A. x ph 
 ->  A. y ps )
 
Theoremcbval 2037 Rule used to change bound variables, using implicit substitution. (Contributed by NM, 5-Aug-1993.) (Revised by Mario Carneiro, 3-Oct-2016.)
 |- 
 F/ y ph   &    |-  F/ x ps   &    |-  ( x  =  y  ->  (
 ph 
 <->  ps ) )   =>    |-  ( A. x ph  <->  A. y ps )
 
Theoremcbvex 2038 Rule used to change bound variables, using implicit substitution. (Contributed by NM, 5-Aug-1993.)
 |- 
 F/ y ph   &    |-  F/ x ps   &    |-  ( x  =  y  ->  (
 ph 
 <->  ps ) )   =>    |-  ( E. x ph  <->  E. y ps )
 
Theoremchvar 2039 Implicit substitution of  y for  x into a theorem. (Contributed by Raph Levien, 9-Jul-2003.) (Revised by Mario Carneiro, 3-Oct-2016.)
 |- 
 F/ x ps   &    |-  ( x  =  y  ->  (
 ph 
 <->  ps ) )   &    |-  ph   =>    |- 
 ps
 
Theoremequvini 2040 A variable introduction law for equality. Lemma 15 of [Monk2] p. 109, however we do not require  z to be distinct from  x and  y (making the proof longer). (Contributed by NM, 5-Aug-1993.) (Proof shortened by Andrew Salmon, 25-May-2011.) (Proof shortened by Wolf Lammen, 7-Apr-2018.)
 |-  ( x  =  y 
 ->  E. z ( x  =  z  /\  z  =  y ) )
 
TheoremequviniOLD 2041 Obsolete proof of equvini 2040 as of 7-Apr-2018. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Andrew Salmon, 25-May-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  ( x  =  y 
 ->  E. z ( x  =  z  /\  z  =  y ) )
 
Theoremequveli 2042 A variable elimination law for equality with no distinct variable requirements. (Compare equvini 2040.) (Contributed by NM, 1-Mar-2013.) (Proof shortened by Mario Carneiro, 17-Oct-2016.) (Proof shortened by Wolf Lammen, 15-Apr-2018.)
 |-  ( A. z ( z  =  x  <->  z  =  y
 )  ->  x  =  y )
 
TheoremequveliOLD 2043 Obsolete proof of equveli 2042 as of 15-Apr-2018. (Contributed by NM, 1-Mar-2013.) (Proof shortened by Mario Carneiro, 17-Oct-2016.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  ( A. z ( z  =  x  <->  z  =  y
 )  ->  x  =  y )
 
Theoremequs45f 2044 Two ways of expressing substitution when  y is not free in  ph. (Contributed by NM, 25-Apr-2008.) (Revised by Mario Carneiro, 4-Oct-2016.)
 |- 
 F/ y ph   =>    |-  ( E. x ( x  =  y  /\  ph )  <->  A. x ( x  =  y  ->  ph )
 )
 
Theoremax11v2 2045* Recovery of ax-11o 2191 from ax11v 2145. This proof uses ax-10 2190 and ax-11 1757. TODO: figure out if this is useful, or if it should be simplified or eliminated. (Contributed by NM, 2-Feb-2007.)
 |-  ( x  =  z 
 ->  ( ph  ->  A. x ( x  =  z  -> 
 ph ) ) )   =>    |-  ( -.  A. x  x  =  y  ->  ( x  =  y  ->  (
 ph  ->  A. x ( x  =  y  ->  ph )
 ) ) )
 
Theoremax11a2 2046* Derive ax-11o 2191 from a hypothesis in the form of ax-11 1757. ax-10 2190 and ax-11 1757 are used by the proof, but not ax-10o 2189 or ax-11o 2191. TODO: figure out if this is useful, or if it should be simplified or eliminated. (Contributed by NM, 2-Feb-2007.)
 |-  ( x  =  z 
 ->  ( A. z ph  ->  A. x ( x  =  z  ->  ph )
 ) )   =>    |-  ( -.  A. x  x  =  y  ->  ( x  =  y  ->  ( ph  ->  A. x ( x  =  y  ->  ph ) ) ) )
 
Theoremax11o 2047 Derivation of set.mm's original ax-11o 2191 from ax-10 2190 and the shorter ax-11 1757 that has replaced it.

An open problem is whether this theorem can be proved without relying on ax-16 2194 or ax-17 1623 (given all of the original and new versions of sp 1759 through ax-15 2193).

Another open problem is whether this theorem can be proved without relying on ax12o 1976.

Theorem ax11 2205 shows the reverse derivation of ax-11 1757 from ax-11o 2191.

Normally, ax11o 2047 should be used rather than ax-11o 2191, except by theorems specifically studying the latter's properties. (Contributed by NM, 3-Feb-2007.)

 |-  ( -.  A. x  x  =  y  ->  ( x  =  y  ->  ( ph  ->  A. x ( x  =  y  ->  ph ) ) ) )
 
Theoremax11b 2048 A bidirectional version of ax11o 2047. (Contributed by NM, 30-Jun-2006.)
 |-  ( ( -.  A. x  x  =  y  /\  x  =  y
 )  ->  ( ph  <->  A. x ( x  =  y  ->  ph ) ) )
 
Theoremequs5 2049 Lemma used in proofs of substitution properties. (Contributed by NM, 5-Aug-1993.)
 |-  ( -.  A. x  x  =  y  ->  ( E. x ( x  =  y  /\  ph )  ->  A. x ( x  =  y  ->  ph )
 ) )
 
Theoremdvelimf 2050 Version of dvelimv 2017 without any variable restrictions. (Contributed by NM, 1-Oct-2002.) (Revised by Mario Carneiro, 6-Oct-2016.)
 |- 
 F/ x ph   &    |-  F/ z ps   &    |-  ( z  =  y  ->  ( ph  <->  ps ) )   =>    |-  ( -.  A. x  x  =  y  ->  F/ x ps )
 
Theoremequvin 2051* A variable introduction law for equality. Lemma 15 of [Monk2] p. 109. (Contributed by NM, 5-Aug-1993.)
 |-  ( x  =  y  <->  E. z ( x  =  z  /\  z  =  y ) )
 
Theoremcbvalv 2052* Rule used to change bound variables, using implicit substitution. (Contributed by NM, 5-Aug-1993.)
 |-  ( x  =  y 
 ->  ( ph  <->  ps ) )   =>    |-  ( A. x ph  <->  A. y ps )
 
Theoremcbvexv 2053* Rule used to change bound variables, using implicit substitution. (Contributed by NM, 5-Aug-1993.)
 |-  ( x  =  y 
 ->  ( ph  <->  ps ) )   =>    |-  ( E. x ph  <->  E. y ps )
 
Theoremcbval2 2054* Rule used to change bound variables, using implicit substitution. (Contributed by NM, 22-Dec-2003.) (Revised by Mario Carneiro, 6-Oct-2016.)
 |- 
 F/ z ph   &    |-  F/ w ph   &    |-  F/ x ps   &    |-  F/ y ps   &    |-  ( ( x  =  z  /\  y  =  w )  ->  ( ph 
 <->  ps ) )   =>    |-  ( A. x A. y ph  <->  A. z A. w ps )
 
Theoremcbvex2 2055* Rule used to change bound variables, using implicit substitution. (Contributed by NM, 14-Sep-2003.) (Revised by Mario Carneiro, 6-Oct-2016.)
 |- 
 F/ z ph   &    |-  F/ w ph   &    |-  F/ x ps   &    |-  F/ y ps   &    |-  ( ( x  =  z  /\  y  =  w )  ->  ( ph 
 <->  ps ) )   =>    |-  ( E. x E. y ph  <->  E. z E. w ps )
 
Theoremcbval2v 2056* Rule used to change bound variables, using implicit substitution. (Contributed by NM, 4-Feb-2005.)
 |-  ( ( x  =  z  /\  y  =  w )  ->  ( ph 
 <->  ps ) )   =>    |-  ( A. x A. y ph  <->  A. z A. w ps )
 
Theoremcbvex2v 2057* Rule used to change bound variables, using implicit substitution. (Contributed by NM, 26-Jul-1995.)
 |-  ( ( x  =  z  /\  y  =  w )  ->  ( ph 
 <->  ps ) )   =>    |-  ( E. x E. y ph  <->  E. z E. w ps )
 
Theoremcbvald 2058* Deduction used to change bound variables, using implicit substitution, particularly useful in conjunction with dvelim 2066. (Contributed by NM, 2-Jan-2002.) (Revised by Mario Carneiro, 6-Oct-2016.)
 |- 
 F/ y ph   &    |-  ( ph  ->  F/ y ps )   &    |-  ( ph  ->  ( x  =  y  ->  ( ps  <->  ch ) ) )   =>    |-  ( ph  ->  (
 A. x ps  <->  A. y ch )
 )
 
Theoremcbvexd 2059* Deduction used to change bound variables, using implicit substitution, particularly useful in conjunction with dvelim 2066. (Contributed by NM, 2-Jan-2002.) (Revised by Mario Carneiro, 6-Oct-2016.)
 |- 
 F/ y ph   &    |-  ( ph  ->  F/ y ps )   &    |-  ( ph  ->  ( x  =  y  ->  ( ps  <->  ch ) ) )   =>    |-  ( ph  ->  ( E. x ps  <->  E. y ch )
 )
 
Theoremcbvaldva 2060* Rule used to change the bound variable in a universal quantifier with implicit substitution. Deduction form. (Contributed by David Moews, 1-May-2017.)
 |-  ( ( ph  /\  x  =  y )  ->  ( ps 
 <->  ch ) )   =>    |-  ( ph  ->  (
 A. x ps  <->  A. y ch )
 )
 
Theoremcbvexdva 2061* Rule used to change the bound variable in an existential quantifier with implicit substitution. Deduction form. (Contributed by David Moews, 1-May-2017.)
 |-  ( ( ph  /\  x  =  y )  ->  ( ps 
 <->  ch ) )   =>    |-  ( ph  ->  ( E. x ps  <->  E. y ch )
 )
 
Theoremcbvex4v 2062* Rule used to change bound variables, using implicit substitution. (Contributed by NM, 26-Jul-1995.)
 |-  ( ( x  =  v  /\  y  =  u )  ->  ( ph 
 <->  ps ) )   &    |-  (
 ( z  =  f 
 /\  w  =  g )  ->  ( ps  <->  ch ) )   =>    |-  ( E. x E. y E. z E. w ph  <->  E. v E. u E. f E. g ch )
 
Theoremchvarv 2063* Implicit substitution of  y for  x into a theorem. (Contributed by NM, 20-Apr-1994.)
 |-  ( x  =  y 
 ->  ( ph  <->  ps ) )   &    |-  ph   =>    |- 
 ps
 
Theoremcleljust 2064* When the class variables in definition df-clel 2400 are replaced with set variables, this theorem of predicate calculus is the result. This theorem provides part of the justification for the consistency of that definition, which "overloads" the set variables in wel 1722 with the class variables in wcel 1721. Note: This proof is referenced on the Metamath Proof Explorer Home Page and shouldn't be changed. (Contributed by NM, 28-Jan-2004.) (Proof modification is discouraged.)
 |-  ( x  e.  y  <->  E. z ( z  =  x  /\  z  e.  y ) )
 
TheoremcleljustALT 2065* When the class variables in definition df-clel 2400 are replaced with set variables, this theorem of predicate calculus is the result. This theorem provides part of the justification for the consistency of that definition, which "overloads" the set variables in wel 1722 with the class variables in wcel 1721. (Contributed by NM, 28-Jan-2004.) (Revised by Mario Carneiro, 21-Dec-2016.) (Proof modification is discouraged.)
 |-  ( x  e.  y  <->  E. z ( z  =  x  /\  z  e.  y ) )
 
Theoremdvelim 2066* This theorem can be used to eliminate a distinct variable restriction on  x and  z and replace it with the "distinctor"  -.  A. x x  =  y as an antecedent.  ph normally has  z free and can be read  ph ( z ), and  ps substitutes  y for  z and can be read  ph ( y ). We don't require that 
x and  y be distinct: if they aren't, the distinctor will become false (in multiple-element domains of discourse) and "protect" the consequent.

To obtain a closed-theorem form of this inference, prefix the hypotheses with  A. x A. z, conjoin them, and apply dvelimdf 2131.

Other variants of this theorem are dvelimh 2015 (with no distinct variable restrictions), dvelimhw 1872 (that avoids ax-12 1946), and dvelimALT 2183 (that avoids ax-10 2190). (Contributed by NM, 23-Nov-1994.)

 |-  ( ph  ->  A. x ph )   &    |-  ( z  =  y  ->  ( ph  <->  ps ) )   =>    |-  ( -.  A. x  x  =  y  ->  ( ps  ->  A. x ps ) )
 
Theoremdvelimnf 2067* Version of dvelim 2066 using "not free" notation. (Contributed by Mario Carneiro, 9-Oct-2016.)
 |- 
 F/ x ph   &    |-  ( z  =  y  ->  ( ph  <->  ps ) )   =>    |-  ( -.  A. x  x  =  y  ->  F/ x ps )
 
Theoremdveel1 2068* Quantifier introduction when one pair of variables is distinct. (Contributed by NM, 2-Jan-2002.)
 |-  ( -.  A. x  x  =  y  ->  ( y  e.  z  ->  A. x  y  e.  z ) )
 
Theoremdveel2 2069* Quantifier introduction when one pair of variables is distinct. (Contributed by NM, 2-Jan-2002.)
 |-  ( -.  A. x  x  =  y  ->  ( z  e.  y  ->  A. x  z  e.  y ) )
 
Theoremax15 2070 Axiom ax-15 2193 is redundant if we assume ax-17 1623. Remark 9.6 in [Megill] p. 448 (p. 16 of the preprint), regarding axiom scheme C14'.

Note that  w is a dummy variable introduced in the proof. On the web page, it is implicitly assumed to be distinct from all other variables. (This is made explicit in the database file set.mm). Its purpose is to satisfy the distinct variable requirements of dveel2 2069 and ax-17 1623. By the end of the proof it has vanished, and the final theorem has no distinct variable requirements. (Contributed by NM, 29-Jun-1995.) (Proof modification is discouraged.)

 |-  ( -.  A. z  z  =  x  ->  ( -.  A. z  z  =  y  ->  ( x  e.  y  ->  A. z  x  e.  y
 ) ) )
 
Theoremdrsb1 2071 Formula-building lemma for use with the Distinctor Reduction Theorem. Part of Theorem 9.4 of [Megill] p. 448 (p. 16 of preprint). (Contributed by NM, 5-Aug-1993.)
 |-  ( A. x  x  =  y  ->  ( [ z  /  x ] ph  <->  [ z  /  y ] ph ) )
 
Theoremsb2 2072 One direction of a simplified definition of substitution. (Contributed by NM, 5-Aug-1993.)
 |-  ( A. x ( x  =  y  ->  ph )  ->  [ y  /  x ] ph )
 
Theoremstdpc4 2073 The specialization axiom of standard predicate calculus. It states that if a statement  ph holds for all  x, then it also holds for the specific case of  y (properly) substituted for  x. Translated to traditional notation, it can be read: " A. x ph ( x )  ->  ph ( y ), provided that  y is free for  x in  ph (
x )." Axiom 4 of [Mendelson] p. 69. See also spsbc 3133 and rspsbc 3199. (Contributed by NM, 5-Aug-1993.)
 |-  ( A. x ph  ->  [ y  /  x ] ph )
 
Theoremsbft 2074 Substitution has no effect on a non-free variable. (Contributed by NM, 30-May-2009.) (Revised by Mario Carneiro, 12-Oct-2016.)
 |-  ( F/ x ph  ->  ( [ y  /  x ] ph  <->  ph ) )
 
Theoremsbf 2075 Substitution for a variable not free in a wff does not affect it. (Contributed by NM, 5-Aug-1993.) (Revised by Mario Carneiro, 4-Oct-2016.)
 |- 
 F/ x ph   =>    |-  ( [ y  /  x ] ph  <->  ph )
 
Theoremsbh 2076 Substitution for a variable not free in a wff does not affect it. (Contributed by NM, 5-Aug-1993.)
 |-  ( ph  ->  A. x ph )   =>    |-  ( [ y  /  x ] ph  <->  ph )
 
Theoremsbf2 2077 Substitution has no effect on a bound variable. (Contributed by NM, 1-Jul-2005.)
 |-  ( [ y  /  x ] A. x ph  <->  A. x ph )
 
Theoremsb6x 2078 Equivalence involving substitution for a variable not free. (Contributed by NM, 5-Aug-1993.) (Revised by Mario Carneiro, 4-Oct-2016.)
 |- 
 F/ x ph   =>    |-  ( [ y  /  x ] ph  <->  A. x ( x  =  y  ->  ph )
 )
 
Theoremnfs1f 2079 If  x is not free in  ph, it is not free in  [ y  /  x ] ph. (Contributed by Mario Carneiro, 11-Aug-2016.)
 |- 
 F/ x ph   =>    |- 
 F/ x [ y  /  x ] ph
 
Theoremsbequ5 2080 Substitution does not change an identical variable specifier. (Contributed by NM, 5-Aug-1993.)
 |-  ( [ w  /  z ] A. x  x  =  y  <->  A. x  x  =  y )
 
Theoremsbequ6 2081 Substitution does not change a distinctor. (Contributed by NM, 5-Aug-1993.)
 |-  ( [ w  /  z ]  -.  A. x  x  =  y  <->  -.  A. x  x  =  y )
 
Theoremsbt 2082 A substitution into a theorem remains true. (See chvar 2039 and chvarv 2063 for versions using implicit substitution.) (Contributed by NM, 21-Jan-2004.) (Proof shortened by Andrew Salmon, 25-May-2011.)
 |-  ph   =>    |- 
 [ y  /  x ] ph
 
Theoremequsb1 2083 Substitution applied to an atomic wff. (Contributed by NM, 5-Aug-1993.)
 |- 
 [ y  /  x ] x  =  y
 
Theoremequsb2 2084 Substitution applied to an atomic wff. (Contributed by NM, 5-Aug-1993.)
 |- 
 [ y  /  x ] y  =  x
 
Theoremsbied 2085 Conversion of implicit substitution to explicit substitution (deduction version of sbie 2087). (Contributed by NM, 30-Jun-1994.) (Revised by Mario Carneiro, 4-Oct-2016.)
 |- 
 F/ x ph   &    |-  ( ph  ->  F/ x ch )   &    |-  ( ph  ->  ( x  =  y  ->  ( ps  <->  ch ) ) )   =>    |-  ( ph  ->  ( [ y  /  x ] ps  <->  ch ) )
 
Theoremsbiedv 2086* Conversion of implicit substitution to explicit substitution (deduction version of sbie 2087). (Contributed by NM, 7-Jan-2017.)
 |-  ( ( ph  /\  x  =  y )  ->  ( ps 
 <->  ch ) )   =>    |-  ( ph  ->  ( [ y  /  x ] ps  <->  ch ) )
 
Theoremsbie 2087 Conversion of implicit substitution to explicit substitution. (Contributed by NM, 30-Jun-1994.) (Revised by Mario Carneiro, 4-Oct-2016.)
 |- 
 F/ x ps   &    |-  ( x  =  y  ->  (
 ph 
 <->  ps ) )   =>    |-  ( [ y  /  x ] ph  <->  ps )
 
Theoremsb6f 2088 Equivalence for substitution when  y is not free in  ph. (Contributed by NM, 5-Aug-1993.) (Revised by Mario Carneiro, 4-Oct-2016.)
 |- 
 F/ y ph   =>    |-  ( [ y  /  x ] ph  <->  A. x ( x  =  y  ->  ph )
 )
 
Theoremsb5f 2089 Equivalence for substitution when  y is not free in  ph. (Contributed by NM, 5-Aug-1993.) (Revised by Mario Carneiro, 4-Oct-2016.)
 |- 
 F/ y ph   =>    |-  ( [ y  /  x ] ph  <->  E. x ( x  =  y  /\  ph )
 )
 
Theoremhbsb2a 2090 Special case of a bound-variable hypothesis builder for substitution. (Contributed by NM, 2-Feb-2007.)
 |-  ( [ y  /  x ] A. y ph  ->  A. x [ y  /  x ] ph )
 
Theoremhbsb2e 2091 Special case of a bound-variable hypothesis builder for substitution. (Contributed by NM, 2-Feb-2007.)
 |-  ( [ y  /  x ] ph  ->  A. x [ y  /  x ] E. y ph )
 
Theoremhbsb3 2092 If  y is not free in  ph,  x is not free in  [ y  /  x ] ph. (Contributed by NM, 5-Aug-1993.)
 |-  ( ph  ->  A. y ph )   =>    |-  ( [ y  /  x ] ph  ->  A. x [ y  /  x ] ph )
 
Theoremnfs1 2093 If  y is not free in  ph,  x is not free in  [ y  /  x ] ph. (Contributed by Mario Carneiro, 11-Aug-2016.)
 |- 
 F/ y ph   =>    |- 
 F/ x [ y  /  x ] ph
 
Theoremax16 2094* Proof of older axiom ax-16 2194. (Contributed by NM, 8-Nov-2006.) (Revised by NM, 22-Sep-2017.)
 |-  ( A. x  x  =  y  ->  ( ph  ->  A. x ph )
 )
 
Theoremax16i 2095* Inference with ax16 2094 as its conclusion. (Contributed by NM, 20-May-2008.) (Proof modification is discouraged.)
 |-  ( x  =  z 
 ->  ( ph  <->  ps ) )   &    |-  ( ps  ->  A. x ps )   =>    |-  ( A. x  x  =  y  ->  ( ph  ->  A. x ph ) )
 
Theoremax16ALT 2096* Alternate proof of ax16 2094. (Contributed by NM, 17-May-2008.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  ( A. x  x  =  y  ->  ( ph  ->  A. x ph )
 )
 
Theoremax16ALT2 2097* Alternate proof of ax16 2094. (Contributed by NM, 8-Nov-2006.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  ( A. x  x  =  y  ->  ( ph  ->  A. x ph )
 )
 
Theorema16gALT 2098* A generalization of axiom ax-16 2194. Alternate proof of a16g 2012 that uses df-sb 1656. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Andrew Salmon, 25-May-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  ( A. x  x  =  y  ->  ( ph  ->  A. z ph )
 )
 
Theorema16gb 2099* A generalization of axiom ax-16 2194. (Contributed by NM, 5-Aug-1993.)
 |-  ( A. x  x  =  y  ->  ( ph 
 <-> 
 A. z ph )
 )
 
Theorema16nf 2100* If dtru 4350 is false, then there is only one element in the universe, so everything satisfies  F/. (Contributed by Mario Carneiro, 7-Oct-2016.)
 |-  ( A. x  x  =  y  ->  F/ z ph )
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