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Theorem List for Metamath Proof Explorer - 2101-2200   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremax15 2101 Axiom ax-15 2219 is redundant if we assume ax-17 1626. 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 2100 and ax-17 1626. 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 2102 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 ) )
 
Theoremsbft 2103 Substitution has no effect on a non-free variable. (Contributed by NM, 30-May-2009.) (Revised by Mario Carneiro, 12-Oct-2016.) (Proof shortened by Wolf Lammen, 3-May-2018.)
 |-  ( F/ x ph  ->  ( [ y  /  x ] ph  <->  ph ) )
 
TheoremsbftOLD 2104 Obsolete proof of sbft 2103 as of 22-Apr-2018. (Contributed by NM, 30-May-2009.) (Revised by Mario Carneiro, 12-Oct-2016.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  ( F/ x ph  ->  ( [ y  /  x ] ph  <->  ph ) )
 
Theoremsbf 2105 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 2106 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 2107 Substitution has no effect on a bound variable. (Contributed by NM, 1-Jul-2005.)
 |-  ( [ y  /  x ] A. x ph  <->  A. x ph )
 
Theoremsb6x 2108 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 2109 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 2110 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 2111 Substitution does not change a distinctor. (Contributed by NM, 5-Aug-1993.)
 |-  ( [ w  /  z ]  -.  A. x  x  =  y  <->  -.  A. x  x  =  y )
 
Theoremsbt 2112 A substitution into a theorem remains true. (See chvar 1968 and chvarv 1969 for versions using implicit substitution.) (Contributed by NM, 21-Jan-2004.) (Proof shortened by Andrew Salmon, 25-May-2011.)
 |-  ph   =>    |- 
 [ y  /  x ] ph
 
Theoremequsb1 2113 Substitution applied to an atomic wff. (Contributed by NM, 5-Aug-1993.)
 |- 
 [ y  /  x ] x  =  y
 
Theoremequsb2 2114 Substitution applied to an atomic wff. (Contributed by NM, 5-Aug-1993.)
 |- 
 [ y  /  x ] y  =  x
 
Theoremdfsb2 2115 An alternate definition of proper substitution that, like df-sb 1659, 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 2116 An alternate definition of proper substitution df-sb 1659 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 )
 ) )
 
Theoremsbn 2117 Negation inside and outside of substitution are equivalent. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Wolf Lammen, 30-Apr-2018.)
 |-  ( [ y  /  x ]  -.  ph  <->  -.  [ y  /  x ] ph )
 
TheoremsbnOLD 2118 Obsolete proof of sbn 2117 as of 30-Apr-2018. (Contributed by NM, 5-Aug-1993.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  ( [ y  /  x ]  -.  ph  <->  -.  [ y  /  x ] ph )
 
Theoremsbi1 2119 Removal of implication from substitution. (Contributed by NM, 5-Aug-1993.)
 |-  ( [ y  /  x ] ( ph  ->  ps )  ->  ( [
 y  /  x ] ph  ->  [ y  /  x ] ps ) )
 
Theoremsbi2 2120 Introduction of implication into substitution. (Contributed by NM, 5-Aug-1993.)
 |-  ( ( [ y  /  x ] ph  ->  [ y  /  x ] ps )  ->  [ y  /  x ] ( ph  ->  ps ) )
 
Theoremsbim 2121 Implication inside and outside of substitution are equivalent. (Contributed by NM, 5-Aug-1993.)
 |-  ( [ y  /  x ] ( ph  ->  ps )  <->  ( [ y  /  x ] ph  ->  [ y  /  x ] ps ) )
 
Theoremsbie 2122 Conversion of implicit substitution to explicit substitution. (Contributed by NM, 30-Jun-1994.) (Revised by Mario Carneiro, 4-Oct-2016.) (Revised by Wolf Lammen, 30-Apr-2018.)
 |- 
 F/ x ps   &    |-  ( x  =  y  ->  (
 ph 
 <->  ps ) )   =>    |-  ( [ y  /  x ] ph  <->  ps )
 
Theoremsbied 2123 Conversion of implicit substitution to explicit substitution (deduction version of sbie 2122). (Contributed by NM, 30-Jun-1994.) (Revised by Mario Carneiro, 4-Oct-2016.) (Proof shortened by Wolf Lammen, 30-Apr-2018.)
 |- 
 F/ x ph   &    |-  ( ph  ->  F/ x ch )   &    |-  ( ph  ->  ( x  =  y  ->  ( ps  <->  ch ) ) )   =>    |-  ( ph  ->  ( [ y  /  x ] ps  <->  ch ) )
 
TheoremsbiedOLD 2124 Obsolete proof of sbied 2123 as of 30-Apr-2018. (Contributed by NM, 30-Jun-1994.) (Revised by Mario Carneiro, 4-Oct-2016.) (Proof modification is discouraged.) (New usage is discouraged.)
 |- 
 F/ x ph   &    |-  ( ph  ->  F/ x ch )   &    |-  ( ph  ->  ( x  =  y  ->  ( ps  <->  ch ) ) )   =>    |-  ( ph  ->  ( [ y  /  x ] ps  <->  ch ) )
 
TheoremsbieOLD 2125 Obsolete proof of sbie 2122 as of 30-Apr-2018. (Contributed by NM, 30-Jun-1994.) (Revised by Mario Carneiro, 4-Oct-2016.) (Proof modification is discouraged.) (New usage is discouraged.)
 |- 
 F/ x ps   &    |-  ( x  =  y  ->  (
 ph 
 <->  ps ) )   =>    |-  ( [ y  /  x ] ph  <->  ps )
 
Theoremsbiedv 2126* Conversion of implicit substitution to explicit substitution (deduction version of sbie 2122). (Contributed by NM, 7-Jan-2017.)
 |-  ( ( ph  /\  x  =  y )  ->  ( ps 
 <->  ch ) )   =>    |-  ( ph  ->  ( [ y  /  x ] ps  <->  ch ) )
 
Theoremsb6f 2127 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 2128 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 )
 )
 
Theoremax16 2129* Proof of older axiom ax-16 2220. (Contributed by NM, 8-Nov-2006.) (Revised by NM, 22-Sep-2017.)
 |-  ( A. x  x  =  y  ->  ( ph  ->  A. x ph )
 )
 
Theoremax16i 2130* Inference with ax16 2129 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 2131* Alternate proof of ax16 2129. (Contributed by NM, 17-May-2008.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  ( A. x  x  =  y  ->  ( ph  ->  A. x ph )
 )
 
Theoremax16ALT2 2132* Alternate proof of ax16 2129. (Contributed by NM, 8-Nov-2006.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  ( A. x  x  =  y  ->  ( ph  ->  A. x ph )
 )
 
Theorema16gALT 2133* A generalization of axiom ax-16 2220. Alternate proof of a16g 2048 that uses df-sb 1659. (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 2134* A generalization of axiom ax-16 2220. (Contributed by NM, 5-Aug-1993.)
 |-  ( A. x  x  =  y  ->  ( ph 
 <-> 
 A. z ph )
 )
 
Theorema16nf 2135* If dtru 4382 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 )
 
Theoremsbequi 2136 An equality theorem for substitution. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Wolf Lammen, 12-May-2018.)
 |-  ( x  =  y 
 ->  ( [ x  /  z ] ph  ->  [ y  /  z ] ph )
 )
 
TheoremsbequiOLD 2137 Obsolete proof of sbequi 2136 as of 2-May-2018. (Contributed by NM, 5-Aug-1993.) (Proof modification is discoraged.) (New usage is discouraged.)
 |-  ( x  =  y 
 ->  ( [ x  /  z ] ph  ->  [ y  /  z ] ph )
 )
 
Theoremsbequ 2138 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 2139 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 ) )
 
Theoremsbor 2140 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 2141 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 2142 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 2143 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 2144 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 2145 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 2146 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 2147 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 2148 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 ) )
 
TheoremspsbeOLD 2149 Obsolete proof of spsbe 1663 as of 3-May-2018. (Contributed by NM, 5-Aug-1993.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  ( [ y  /  x ] ph  ->  E. x ph )
 
Theoremspsbim 2150 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 2151 Specialization of biconditional. (Contributed by NM, 5-Aug-1993.)
 |-  ( A. x (
 ph 
 <->  ps )  ->  ( [ y  /  x ] ph  <->  [ y  /  x ] ps ) )
 
Theoremsbbid 2152 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 2153 Elimination of equality from antecedent after substitution. (Contributed by NM, 5-Aug-1993.)
 |-  ( [ y  /  x ] ph  <->  [ y  /  x ] ( x  =  y  ->  ph ) )
 
Theoremnfsb4t 2154 A variable not free remains so after substitution with a distinct variable (closed form of nfsb4 2156). (Contributed by NM, 7-Apr-2004.) (Revised by Mario Carneiro, 4-Oct-2016.) (Proof shortened by Wolf Lammen, 11-May-2018.)
 |-  ( A. x F/ z ph  ->  ( -.  A. z  z  =  y  ->  F/ z [ y  /  x ] ph ) )
 
Theoremnfsb4tOLD 2155 Obsolete proof of nfsb4t 2154 as of 6-May-2018. (Contributed by NM, 7-Apr-2004.) (Revised by Mario Carneiro, 4-Oct-2016.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  ( A. x F/ z ph  ->  ( -.  A. z  z  =  y  ->  F/ z [ y  /  x ] ph ) )
 
Theoremnfsb4 2156 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 )
 
TheoremdvelimdfOLD 2157 Obsolete proof of dvelimdf 2066 as of 6-May-2018. (Contributed by NM, 7-Apr-2004.) (Revised by Mario Carneiro, 6-Oct-2016.) (Proof modification is discouraged.) (New usage is discouraged.)
 |- 
 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 2158 A composition law for substitution. (Contributed by NM, 5-Aug-1993.)
 |-  ( [ y  /  x ] [ x  /  y ] ph  <->  [ y  /  x ] ph )
 
Theoremsbid2 2159 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 2160 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 2161 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 2162 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 2163 A composition law for substitution. (Contributed by NM, 5-Aug-1993.)
 |-  ( [ z  /  y ] [ y  /  x ] ph  <->  [ z  /  x ] [ x  /  y ] ph )
 
Theoremsbcom 2164 A commutativity law for substitution. (Contributed by NM, 27-May-1997.)
 |-  ( [ y  /  z ] [ y  /  x ] ph  <->  [ y  /  x ] [ y  /  z ] ph )
 
Theoremsb5rf 2165 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 2166 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 2167 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 2168 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 2169 Commutation of quantification and substitution variables. (Contributed by NM, 5-Aug-1993.)
 |-  ( A. x [ x  /  y ] ph  ->  A. y [ y  /  x ] ph )
 
Theoremsb9 2170 Commutation of quantification and substitution variables. (Contributed by NM, 5-Aug-1993.)
 |-  ( A. x [ x  /  y ] ph  <->  A. y [ y  /  x ] ph )
 
Theoremax11v 2171* This is a version of ax-11o 2217 when the variables are distinct. Axiom (C8) of [Monk2] p. 105. See theorem ax11v2 2074 for the rederivation of ax-11o 2217 from this theorem. (Contributed by NM, 5-Aug-1993.)
 |-  ( x  =  y 
 ->  ( ph  ->  A. x ( x  =  y  -> 
 ph ) ) )
 
Theoremax11vALT 2172* Alternate proof of ax11v 2171 that avoids theorem ax16 2129 and is proved directly from ax-11 1761 rather than via ax11o 2077. (Contributed by Jim Kingdon, 15-Dec-2017.) (New usage is discouraged.) (Proof modification is discouraged.)
 |-  ( x  =  y 
 ->  ( ph  ->  A. x ( x  =  y  -> 
 ph ) ) )
 
Theoremsb56 2173* Two equivalent ways of expressing the proper substitution of  y for  x in  ph, when  x and  y are distinct. Theorem 6.2 of [Quine] p. 40. The proof does not involve df-sb 1659. (Contributed by NM, 14-Apr-2008.)
 |-  ( E. x ( x  =  y  /\  ph )  <->  A. x ( x  =  y  ->  ph )
 )
 
Theoremsb6 2174* Equivalence for substitution. Compare Theorem 6.2 of [Quine] p. 40. Also proved as Lemmas 16 and 17 of [Tarski] p. 70. (Contributed by NM, 18-Aug-1993.)
 |-  ( [ y  /  x ] ph  <->  A. x ( x  =  y  ->  ph )
 )
 
Theoremsb5 2175* Equivalence for substitution. Similar to Theorem 6.1 of [Quine] p. 40. (Contributed by NM, 18-Aug-1993.)
 |-  ( [ y  /  x ] ph  <->  E. x ( x  =  y  /\  ph )
 )
 
Theoremequsb3lem 2176* Lemma for equsb3 2177. (Contributed by Raph Levien and FL, 4-Dec-2005.) (Proof shortened by Andrew Salmon, 14-Jun-2011.)
 |-  ( [ x  /  y ] y  =  z  <-> 
 x  =  z )
 
Theoremequsb3 2177* Substitution applied to an atomic wff. (Contributed by Raph Levien and FL, 4-Dec-2005.)
 |-  ( [ x  /  y ] y  =  z  <-> 
 x  =  z )
 
Theoremelsb3 2178* Substitution applied to an atomic membership wff. (Contributed by NM, 7-Nov-2006.) (Proof shortened by Andrew Salmon, 14-Jun-2011.)
 |-  ( [ x  /  y ] y  e.  z  <->  x  e.  z )
 
Theoremelsb4 2179* Substitution applied to an atomic membership wff. (Contributed by Rodolfo Medina, 3-Apr-2010.) (Proof shortened by Andrew Salmon, 14-Jun-2011.)
 |-  ( [ x  /  y ] z  e.  y  <->  z  e.  x )
 
Theoremhbs1 2180*  x is not free in  [ y  /  x ] ph when  x and  y are distinct. (Contributed by NM, 5-Aug-1993.)
 |-  ( [ y  /  x ] ph  ->  A. x [ y  /  x ] ph )
 
Theoremnfs1v 2181*  x is not free in  [ y  /  x ] ph when  x and  y are distinct. (Contributed by Mario Carneiro, 11-Aug-2016.)
 |- 
 F/ x [ y  /  x ] ph
 
Theoremsbhb 2182* Two ways of expressing " x is (effectively) not free in  ph." (Contributed by NM, 29-May-2009.)
 |-  ( ( ph  ->  A. x ph )  <->  A. y ( ph  ->  [ y  /  x ] ph ) )
 
Theoremsbnf2 2183* Two ways of expressing " x is (effectively) not free in  ph." (Contributed by Gérard Lang, 14-Nov-2013.) (Revised by Mario Carneiro, 6-Oct-2016.)
 |-  ( F/ x ph  <->  A. y A. z ( [
 y  /  x ] ph 
 <->  [ z  /  x ] ph ) )
 
Theoremnfsb 2184* If  z is not free in  ph, it is not free in  [ y  /  x ] ph when  y and  z are distinct. (Contributed by Mario Carneiro, 11-Aug-2016.)
 |- 
 F/ z ph   =>    |- 
 F/ z [ y  /  x ] ph
 
Theoremhbsb 2185* If  z is not free in  ph, it is not free in  [ y  /  x ] ph when  y and  z are distinct. (Contributed by NM, 12-Aug-1993.)
 |-  ( ph  ->  A. z ph )   =>    |-  ( [ y  /  x ] ph  ->  A. z [ y  /  x ] ph )
 
Theoremnfsbd 2186* Deduction version of nfsb 2184. (Contributed by NM, 15-Feb-2013.)
 |- 
 F/ x ph   &    |-  ( ph  ->  F/ z ps )   =>    |-  ( ph  ->  F/ z [ y  /  x ] ps )
 
Theorem2sb5 2187* Equivalence for double substitution. (Contributed by NM, 3-Feb-2005.)
 |-  ( [ z  /  x ] [ w  /  y ] ph  <->  E. x E. y
 ( ( x  =  z  /\  y  =  w )  /\  ph )
 )
 
Theorem2sb6 2188* Equivalence for double substitution. (Contributed by NM, 3-Feb-2005.)
 |-  ( [ z  /  x ] [ w  /  y ] ph  <->  A. x A. y
 ( ( x  =  z  /\  y  =  w )  ->  ph )
 )
 
Theoremsbcom2 2189* Commutativity law for substitution. Used in proof of Theorem 9.7 of [Megill] p. 449 (p. 16 of the preprint). (Contributed by NM, 27-May-1997.)
 |-  ( [ w  /  z ] [ y  /  x ] ph  <->  [ y  /  x ] [ w  /  z ] ph )
 
Theorempm11.07 2190* (Probably not) Axiom *11.07 in [WhiteheadRussell] p. 159. The original confusingly reads: *11.07 "Whatever possible argument  x may be,  ph ( x ,  y ) is true whatever possible argument  y may be" implies the corresponding statement with  x and  y interchanged except in " ph ( x ,  y )". This theorem will be deleted after 22-Feb-2018 if no one is able to determine the correct interpretation. See https://groups.google.com/d/msg/metamath/iS0fOvSemC8/YzrRyX70AgAJ. (Contributed by Andrew Salmon, 17-Jun-2011.) (Proof shortened by Jim Kingdon, 22-Jan-2018.) (New usage is discouraged.)
 |-  ( [ w  /  x ] [ y  /  z ] ph  <->  [ y  /  x ] [ w  /  z ] ph )
 
Theorempm11.07OLD 2191* Obsolete proof of pm11.07 2190 as of 22-Jan-2018. (Contributed by Andrew Salmon, 17-Jun-2011.) (New usage is discouraged.) (Proof modification is discouraged.)
 |-  ( [ w  /  x ] [ y  /  z ] ph  <->  [ y  /  x ] [ w  /  z ] ph )
 
Theoremsb6a 2192* Equivalence for substitution. (Contributed by NM, 5-Aug-1993.)
 |-  ( [ y  /  x ] ph  <->  A. x ( x  =  y  ->  [ x  /  y ] ph )
 )
 
Theorem2sb5rf 2193* Reversed double substitution. (Contributed by NM, 3-Feb-2005.) (Revised by Mario Carneiro, 6-Oct-2016.)
 |- 
 F/ z ph   &    |-  F/ w ph   =>    |-  ( ph 
 <-> 
 E. z E. w ( ( z  =  x  /\  w  =  y )  /\  [
 z  /  x ] [ w  /  y ] ph ) )
 
Theorem2sb6rf 2194* Reversed double substitution. (Contributed by NM, 3-Feb-2005.) (Revised by Mario Carneiro, 6-Oct-2016.)
 |- 
 F/ z ph   &    |-  F/ w ph   =>    |-  ( ph 
 <-> 
 A. z A. w ( ( z  =  x  /\  w  =  y )  ->  [ z  /  x ] [ w  /  y ] ph )
 )
 
Theoremsb7f 2195* This version of dfsb7 2197 does not require that  ph and  z be distinct. This permits it to be used as a definition for substitution in a formalization that omits the logically redundant axiom ax-17 1626 i.e. that doesn't have the concept of a variable not occurring in a wff. (df-sb 1659 is also suitable, but its mixing of free and bound variables is distasteful to some logicians.) (Contributed by NM, 26-Jul-2006.) (Revised by Mario Carneiro, 6-Oct-2016.)
 |- 
 F/ z ph   =>    |-  ( [ y  /  x ] ph  <->  E. z ( z  =  y  /\  E. x ( x  =  z  /\  ph )
 ) )
 
Theoremsb7h 2196* This version of dfsb7 2197 does not require that  ph and  z be distinct. This permits it to be used as a definition for substitution in a formalization that omits the logically redundant axiom ax-17 1626 i.e. that doesn't have the concept of a variable not occurring in a wff. (df-sb 1659 is also suitable, but its mixing of free and bound variables is distasteful to some logicians.) (Contributed by NM, 26-Jul-2006.) (Proof shortened by Andrew Salmon, 25-May-2011.)
 |-  ( ph  ->  A. z ph )   =>    |-  ( [ y  /  x ] ph  <->  E. z ( z  =  y  /\  E. x ( x  =  z  /\  ph )
 ) )
 
Theoremdfsb7 2197* An alternate definition of proper substitution df-sb 1659. By introducing a dummy variable  z in the definiens, we are able to eliminate any distinct variable restrictions among the variables  x,  y, and  ph of the definiendum. No distinct variable conflicts arise because  z effectively insulates  x from  y. To achieve this, we use a chain of two substitutions in the form of sb5 2175, first  z for  x then  y for  z. Compare Definition 2.1'' of [Quine] p. 17, which is obtained from this theorem by applying df-clab 2422. Theorem sb7h 2196 provides a version where  ph and  z don't have to be distinct. (Contributed by NM, 28-Jan-2004.)
 |-  ( [ y  /  x ] ph  <->  E. z ( z  =  y  /\  E. x ( x  =  z  /\  ph )
 ) )
 
Theoremsb10f 2198* Hao Wang's identity axiom P6 in Irving Copi, Symbolic Logic (5th ed., 1979), p. 328. In traditional predicate calculus, this is a sole axiom for identity from which the usual ones can be derived. (Contributed by NM, 9-May-2005.) (Revised by Mario Carneiro, 6-Oct-2016.)
 |- 
 F/ x ph   =>    |-  ( [ y  /  z ] ph  <->  E. x ( x  =  y  /\  [ x  /  z ] ph ) )
 
Theoremsbid2v 2199* An identity law for substitution. Used in proof of Theorem 9.7 of [Megill] p. 449 (p. 16 of the preprint). (Contributed by NM, 5-Aug-1993.)
 |-  ( [ y  /  x ] [ x  /  y ] ph  <->  ph )
 
Theoremsbelx 2200* Elimination of substitution. (Contributed by NM, 5-Aug-1993.)
 |-  ( ph  <->  E. x ( x  =  y  /\  [ x  /  y ] ph ) )
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