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Theorem List for Metamath Proof Explorer - 2001-2100   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremequs5 2001 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 2002 Version of dvelimv 1944 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 )
 
Theoremspv 2003* Specialization, using implicit substitution. (Contributed by NM, 30-Aug-1993.)
 |-  ( x  =  y 
 ->  ( ph  <->  ps ) )   =>    |-  ( A. x ph 
 ->  ps )
 
Theoremspimev 2004* Distinct-variable version of spime 1981. (Contributed by NM, 5-Aug-1993.)
 |-  ( x  =  y 
 ->  ( ph  ->  ps )
 )   =>    |-  ( ph  ->  E. x ps )
 
Theoremspeiv 2005* Inference from existential specialization, using implicit substitution. (Contributed by NM, 19-Aug-1993.)
 |-  ( x  =  y 
 ->  ( ph  <->  ps ) )   &    |-  ps   =>    |-  E. x ph
 
Theoremequvin 2006* 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 2007* 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 2008* 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 2009* 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 2010* 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 2011* 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 2012* 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 2013* Deduction used to change bound variables, using implicit substitution, particularly useful in conjunction with dvelim 2021. (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 2014* Deduction used to change bound variables, using implicit substitution, particularly useful in conjunction with dvelim 2021. (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 2015* 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 2016* 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 2017* 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 2018* Implicit substitution of  y for  x into a theorem. (Contributed by NM, 20-Apr-1994.)
 |-  ( x  =  y 
 ->  ( ph  <->  ps ) )   &    |-  ph   =>    |- 
 ps
 
Theoremcleljust 2019* When the class variables in definition df-clel 2354 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 1711 with the class variables in wcel 1710. 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 2020* When the class variables in definition df-clel 2354 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 1711 with the class variables in wcel 1710. (Contributed by NM, 28-Jan-2004.) (Revised by Mario Carneiro, 21-Dec-2016.)
 |-  ( x  e.  y  <->  E. z ( z  =  x  /\  z  e.  y ) )
 
Theoremdvelim 2021* 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 2087.

Other variants of this theorem are dvelimh 1969 (with no distinct variable restrictions), dvelimhw 1854 (that avoids ax-12 1930), and dvelimALT 2138 (that avoids ax-10 2145). (Contributed by NM, 23-Nov-1994.)

 |-  ( ph  ->  A. x ph )   &    |-  ( z  =  y  ->  ( ph  <->  ps ) )   =>    |-  ( -.  A. x  x  =  y  ->  ( ps  ->  A. x ps ) )
 
Theoremdvelimnf 2022* Version of dvelim 2021 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 )
 
Theoremdveeq1 2023* Quantifier introduction when one pair of variables is distinct. (Contributed by NM, 2-Jan-2002.)
 |-  ( -.  A. x  x  =  y  ->  ( y  =  z  ->  A. x  y  =  z ) )
 
Theoremdveel1 2024* 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 2025* 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 2026 Axiom ax-15 2148 is redundant if we assume ax-17 1616. 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 2025 and ax-17 1616. 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 2027 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 2028 One direction of a simplified definition of substitution. (Contributed by NM, 5-Aug-1993.)
 |-  ( A. x ( x  =  y  ->  ph )  ->  [ y  /  x ] ph )
 
Theoremstdpc4 2029 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 3079 and rspsbc 3145. (Contributed by NM, 5-Aug-1993.)
 |-  ( A. x ph  ->  [ y  /  x ] ph )
 
Theoremsbft 2030 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 2031 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 2032 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 2033 Substitution has no effect on a bound variable. (Contributed by NM, 1-Jul-2005.)
 |-  ( [ y  /  x ] A. x ph  <->  A. x ph )
 
Theoremsb6x 2034 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 2035 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 2036 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 2037 Substitution does not change a distinctor. (Contributed by NM, 5-Aug-1993.)
 |-  ( [ w  /  z ]  -.  A. x  x  =  y  <->  -.  A. x  x  =  y )
 
Theoremsbt 2038 A substitution into a theorem remains true. (See chvar 1991 and chvarv 2018 for versions using implicit substitution.) (Contributed by NM, 21-Jan-2004.) (Proof shortened by Andrew Salmon, 25-May-2011.)
 |-  ph   =>    |- 
 [ y  /  x ] ph
 
Theoremequsb1 2039 Substitution applied to an atomic wff. (Contributed by NM, 5-Aug-1993.)
 |- 
 [ y  /  x ] x  =  y
 
Theoremequsb2 2040 Substitution applied to an atomic wff. (Contributed by NM, 5-Aug-1993.)
 |- 
 [ y  /  x ] y  =  x
 
Theoremsbied 2041 Conversion of implicit substitution to explicit substitution (deduction version of sbie 2043). (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 2042* Conversion of implicit substitution to explicit substitution (deduction version of sbie 2043). (Contributed by NM, 7-Jan-2017.)
 |-  ( ( ph  /\  x  =  y )  ->  ( ps 
 <->  ch ) )   =>    |-  ( ph  ->  ( [ y  /  x ] ps  <->  ch ) )
 
Theoremsbie 2043 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 2044 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 2045 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 2046 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 2047 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 2048 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 2049 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 2050* Proof of older axiom ax-16 2149. (Contributed by NM, 8-Nov-2006.) (Revised by NM, 22-Sep-2017.)
 |-  ( A. x  x  =  y  ->  ( ph  ->  A. x ph )
 )
 
Theoremax16i 2051* Inference with ax16 2050 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 2052* Alternate proof of ax16 2050. (Contributed by NM, 17-May-2008.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  ( A. x  x  =  y  ->  ( ph  ->  A. x ph )
 )
 
Theoremax16ALT2 2053* Alternate proof of ax16 2050. (Contributed by NM, 8-Nov-2006.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  ( A. x  x  =  y  ->  ( ph  ->  A. x ph )
 )
 
Theorema16gALT 2054* A generalization of axiom ax-16 2149. Alternate proof of a16g 1950 that uses df-sb 1649. (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 2055* A generalization of axiom ax-16 2149. (Contributed by NM, 5-Aug-1993.)
 |-  ( A. x  x  =  y  ->  ( ph 
 <-> 
 A. z ph )
 )
 
Theorema16nf 2056* If dtru 4280 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 )
 
Theoremsb3 2057 One direction of a simplified definition of substitution when variables are distinct. (Contributed by NM, 5-Aug-1993.)
 |-  ( -.  A. x  x  =  y  ->  ( E. x ( x  =  y  /\  ph )  ->  [ y  /  x ] ph ) )
 
Theoremsb4 2058 One direction of a simplified definition of substitution when variables are distinct. (Contributed by NM, 5-Aug-1993.)
 |-  ( -.  A. x  x  =  y  ->  ( [ y  /  x ] ph  ->  A. x ( x  =  y  ->  ph ) ) )
 
Theoremsb4b 2059 Simplified definition of substitution when variables are distinct. (Contributed by NM, 27-May-1997.)
 |-  ( -.  A. x  x  =  y  ->  ( [ y  /  x ] ph  <->  A. x ( x  =  y  ->  ph )
 ) )
 
Theoremdfsb2 2060 An alternate definition of proper substitution that, like df-sb 1649, mixes free and bound variables to avoid distinct variable requirements. (Contributed by NM, 17-Feb-2005.)
 |-  ( [ y  /  x ] ph  <->  ( ( x  =  y  /\  ph )  \/  A. x ( x  =  y  ->  ph )
 ) )
 
Theoremdfsb3 2061 An alternate definition of proper substitution df-sb 1649 that uses only primitive connectives (no defined terms) on the right-hand side. (Contributed by NM, 6-Mar-2007.)
 |-  ( [ y  /  x ] ph  <->  ( ( x  =  y  ->  -.  ph )  ->  A. x ( x  =  y  ->  ph )
 ) )
 
Theoremhbsb2 2062 Bound-variable hypothesis builder for substitution. (Contributed by NM, 5-Aug-1993.)
 |-  ( -.  A. x  x  =  y  ->  ( [ y  /  x ] ph  ->  A. x [
 y  /  x ] ph ) )
 
Theoremnfsb2 2063 Bound-variable hypothesis builder for substitution. (Contributed by Mario Carneiro, 4-Oct-2016.)
 |-  ( -.  A. x  x  =  y  ->  F/ x [ y  /  x ] ph )
 
Theoremsbequi 2064 An equality theorem for substitution. (Contributed by NM, 5-Aug-1993.)
 |-  ( x  =  y 
 ->  ( [ x  /  z ] ph  ->  [ y  /  z ] ph )
 )
 
Theoremsbequ 2065 An equality theorem for substitution. Used in proof of Theorem 9.7 in [Megill] p. 449 (p. 16 of the preprint). (Contributed by NM, 5-Aug-1993.)
 |-  ( x  =  y 
 ->  ( [ x  /  z ] ph  <->  [ y  /  z ] ph ) )
 
Theoremdrsb2 2066 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  ->  ( [ x  /  z ] ph  <->  [ y  /  z ] ph ) )
 
Theoremsbn 2067 Negation inside and outside of substitution are equivalent. (Contributed by NM, 5-Aug-1993.)
 |-  ( [ y  /  x ]  -.  ph  <->  -.  [ y  /  x ] ph )
 
Theoremsbi1 2068 Removal of implication from substitution. (Contributed by NM, 5-Aug-1993.)
 |-  ( [ y  /  x ] ( ph  ->  ps )  ->  ( [
 y  /  x ] ph  ->  [ y  /  x ] ps ) )
 
Theoremsbi2 2069 Introduction of implication into substitution. (Contributed by NM, 5-Aug-1993.)
 |-  ( ( [ y  /  x ] ph  ->  [ y  /  x ] ps )  ->  [ y  /  x ] ( ph  ->  ps ) )
 
Theoremsbim 2070 Implication inside and outside of substitution are equivalent. (Contributed by NM, 5-Aug-1993.)
 |-  ( [ y  /  x ] ( ph  ->  ps )  <->  ( [ y  /  x ] ph  ->  [ y  /  x ] ps ) )
 
Theoremsbor 2071 Logical OR inside and outside of substitution are equivalent. (Contributed by NM, 29-Sep-2002.)
 |-  ( [ y  /  x ] ( ph  \/  ps )  <->  ( [ y  /  x ] ph  \/  [ y  /  x ] ps ) )
 
Theoremsbrim 2072 Substitution with a variable not free in antecedent affects only the consequent. (Contributed by NM, 5-Aug-1993.) (Revised by Mario Carneiro, 4-Oct-2016.)
 |- 
 F/ x ph   =>    |-  ( [ y  /  x ] ( ph  ->  ps )  <->  ( ph  ->  [ y  /  x ] ps ) )
 
Theoremsblim 2073 Substitution with a variable not free in consequent affects only the antecedent. (Contributed by NM, 14-Nov-2013.) (Revised by Mario Carneiro, 4-Oct-2016.)
 |- 
 F/ x ps   =>    |-  ( [ y  /  x ] ( ph  ->  ps )  <->  ( [ y  /  x ] ph  ->  ps ) )
 
Theoremsban 2074 Conjunction inside and outside of a substitution are equivalent. (Contributed by NM, 5-Aug-1993.)
 |-  ( [ y  /  x ] ( ph  /\  ps ) 
 <->  ( [ y  /  x ] ph  /\  [
 y  /  x ] ps ) )
 
Theoremsb3an 2075 Conjunction inside and outside of a substitution are equivalent. (Contributed by NM, 14-Dec-2006.)
 |-  ( [ y  /  x ] ( ph  /\  ps  /\ 
 ch )  <->  ( [ y  /  x ] ph  /\  [
 y  /  x ] ps  /\  [ y  /  x ] ch ) )
 
Theoremsbbi 2076 Equivalence inside and outside of a substitution are equivalent. (Contributed by NM, 5-Aug-1993.)
 |-  ( [ y  /  x ] ( ph  <->  ps )  <->  ( [ y  /  x ] ph  <->  [ y  /  x ] ps ) )
 
Theoremsblbis 2077 Introduce left biconditional inside of a substitution. (Contributed by NM, 19-Aug-1993.)
 |-  ( [ y  /  x ] ph  <->  ps )   =>    |-  ( [ y  /  x ] ( ch  <->  ph )  <->  ( [ y  /  x ] ch  <->  ps ) )
 
Theoremsbrbis 2078 Introduce right biconditional inside of a substitution. (Contributed by NM, 18-Aug-1993.)
 |-  ( [ y  /  x ] ph  <->  ps )   =>    |-  ( [ y  /  x ] ( ph  <->  ch )  <->  ( ps  <->  [ y  /  x ] ch ) )
 
Theoremsbrbif 2079 Introduce right biconditional inside of a substitution. (Contributed by NM, 18-Aug-1993.) (Revised by Mario Carneiro, 4-Oct-2016.)
 |- 
 F/ x ch   &    |-  ( [ y  /  x ] ph  <->  ps )   =>    |-  ( [ y  /  x ] ( ph  <->  ch )  <->  ( ps  <->  ch ) )
 
Theoremspsbe 2080 A specialization theorem. (Contributed by NM, 5-Aug-1993.)
 |-  ( [ y  /  x ] ph  ->  E. x ph )
 
Theoremspsbim 2081 Specialization of implication. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Andrew Salmon, 25-May-2011.)
 |-  ( A. x (
 ph  ->  ps )  ->  ( [ y  /  x ] ph  ->  [ y  /  x ] ps )
 )
 
Theoremspsbbi 2082 Specialization of biconditional. (Contributed by NM, 5-Aug-1993.)
 |-  ( A. x (
 ph 
 <->  ps )  ->  ( [ y  /  x ] ph  <->  [ y  /  x ] ps ) )
 
Theoremsbbid 2083 Deduction substituting both sides of a biconditional. (Contributed by NM, 5-Aug-1993.)
 |- 
 F/ x ph   &    |-  ( ph  ->  ( ps  <->  ch ) )   =>    |-  ( ph  ->  ( [ y  /  x ] ps  <->  [ y  /  x ] ch ) )
 
Theoremsbequ8 2084 Elimination of equality from antecedent after substitution. (Contributed by NM, 5-Aug-1993.)
 |-  ( [ y  /  x ] ph  <->  [ y  /  x ] ( x  =  y  ->  ph ) )
 
Theoremnfsb4t 2085 A variable not free remains so after substitution with a distinct variable (closed form of nfsb4 2086). (Contributed by NM, 7-Apr-2004.) (Revised by Mario Carneiro, 4-Oct-2016.)
 |-  ( A. x F/ z ph  ->  ( -.  A. z  z  =  y  ->  F/ z [ y  /  x ] ph ) )
 
Theoremnfsb4 2086 A variable not free remains so after substitution with a distinct variable. (Contributed by NM, 5-Aug-1993.) (Revised by Mario Carneiro, 4-Oct-2016.)
 |- 
 F/ z ph   =>    |-  ( -.  A. z  z  =  y  ->  F/ z [ y  /  x ] ph )
 
Theoremdvelimdf 2087 Deduction form of dvelimf 2002. This version may be useful if we want to avoid ax-17 1616 and use ax-16 2149 instead. (Contributed by NM, 7-Apr-2004.) (Revised by Mario Carneiro, 6-Oct-2016.)
 |- 
 F/ x ph   &    |-  F/ z ph   &    |-  ( ph  ->  F/ x ps )   &    |-  ( ph  ->  F/ z ch )   &    |-  ( ph  ->  ( z  =  y  ->  ( ps  <->  ch ) ) )   =>    |-  ( ph  ->  ( -.  A. x  x  =  y 
 ->  F/ x ch )
 )
 
Theoremsbco 2088 A composition law for substitution. (Contributed by NM, 5-Aug-1993.)
 |-  ( [ y  /  x ] [ x  /  y ] ph  <->  [ y  /  x ] ph )
 
Theoremsbid2 2089 An identity law for substitution. (Contributed by NM, 5-Aug-1993.) (Revised by Mario Carneiro, 6-Oct-2016.)
 |- 
 F/ x ph   =>    |-  ( [ y  /  x ] [ x  /  y ] ph  <->  ph )
 
Theoremsbidm 2090 An idempotent law for substitution. (Contributed by NM, 30-Jun-1994.) (Proof shortened by Andrew Salmon, 25-May-2011.)
 |-  ( [ y  /  x ] [ y  /  x ] ph  <->  [ y  /  x ] ph )
 
Theoremsbco2 2091 A composition law for substitution. (Contributed by NM, 30-Jun-1994.) (Revised by Mario Carneiro, 6-Oct-2016.)
 |- 
 F/ z ph   =>    |-  ( [ y  /  z ] [ z  /  x ] ph  <->  [ y  /  x ] ph )
 
Theoremsbco2d 2092 A composition law for substitution. (Contributed by NM, 5-Aug-1993.) (Revised by Mario Carneiro, 6-Oct-2016.)
 |- 
 F/ x ph   &    |-  F/ z ph   &    |-  ( ph  ->  F/ z ps )   =>    |-  ( ph  ->  ( [ y  /  z ] [ z  /  x ] ps  <->  [ y  /  x ] ps ) )
 
Theoremsbco3 2093 A composition law for substitution. (Contributed by NM, 5-Aug-1993.)
 |-  ( [ z  /  y ] [ y  /  x ] ph  <->  [ z  /  x ] [ x  /  y ] ph )
 
Theoremsbcom 2094 A commutativity law for substitution. (Contributed by NM, 27-May-1997.)
 |-  ( [ y  /  z ] [ y  /  x ] ph  <->  [ y  /  x ] [ y  /  z ] ph )
 
Theoremsb5rf 2095 Reversed substitution. (Contributed by NM, 3-Feb-2005.) (Revised by Mario Carneiro, 6-Oct-2016.)
 |- 
 F/ y ph   =>    |-  ( ph  <->  E. y ( y  =  x  /\  [
 y  /  x ] ph ) )
 
Theoremsb6rf 2096 Reversed substitution. (Contributed by NM, 5-Aug-1993.) (Revised by Mario Carneiro, 6-Oct-2016.)
 |- 
 F/ y ph   =>    |-  ( ph  <->  A. y ( y  =  x  ->  [ y  /  x ] ph )
 )
 
Theoremsb8 2097 Substitution of variable in universal quantifier. (Contributed by NM, 5-Aug-1993.) (Revised by Mario Carneiro, 6-Oct-2016.) (Proof shortened by Jim Kingdon, 15-Jan-2018.)
 |- 
 F/ y ph   =>    |-  ( A. x ph  <->  A. y [ y  /  x ] ph )
 
Theoremsb8e 2098 Substitution of variable in existential quantifier. (Contributed by NM, 12-Aug-1993.) (Revised by Mario Carneiro, 6-Oct-2016.) (Proof shortened by Jim Kingdon, 15-Jan-2018.)
 |- 
 F/ y ph   =>    |-  ( E. x ph  <->  E. y [ y  /  x ] ph )
 
Theoremsb9i 2099 Commutation of quantification and substitution variables. (Contributed by NM, 5-Aug-1993.)
 |-  ( A. x [ x  /  y ] ph  ->  A. y [ y  /  x ] ph )
 
Theoremsb9 2100 Commutation of quantification and substitution variables. (Contributed by NM, 5-Aug-1993.)
 |-  ( A. x [ x  /  y ] ph  <->  A. y [ y  /  x ] ph )
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