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Theorem List for Metamath Proof Explorer - 29201-29300   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremdrnf1NEW7 29201 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 )
 )
 
Theoremdral2wAUX7 29202* Weak version of dral2 2020 not requiring ax-7 1745. (Contributed by NM, 27-Oct-2017.)
 |-  ( A. x  x  =  y  ->  ( ph  <->  ps ) )   =>    |-  ( A. x  x  =  y  ->  (
 A. z ph  <->  A. z ps )
 )
 
Theoremdrex2wAUX7 29203* Weak version of drex2 2025 not requiring ax-7 1745. (Contributed by NM, 27-Oct-2017.)
 |-  ( A. x  x  =  y  ->  ( ph  <->  ps ) )   =>    |-  ( A. x  x  =  y  ->  ( E. z ph  <->  E. z ps )
 )
 
Theoremdrnf2wAUX7 29204* Weak version of drnf2 2027 not requiring ax-7 1745. (Contributed by NM, 27-Oct-2017.)
 |-  ( A. x  x  =  y  ->  ( ph  <->  ps ) )   =>    |-  ( A. x  x  =  y  ->  ( F/ z ph  <->  F/ z ps )
 )
 
Theoremdral2w2AUX7 29205* Weak version of dral2 2020 not requiring ax-7 1745. (Contributed by NM, 25-Nov-2017.)
 |-  ( A. x  x  =  y  ->  ( ph  <->  ps ) )   =>    |-  ( A. x  x  =  y  ->  (
 A. z ph  <->  A. z ps )
 )
 
Theoremdrex2w2AUX7 29206* Weak version of drex2 2025 not requiring ax-7 1745. (Contributed by NM, 25-Nov-2017.)
 |-  ( A. x  x  =  y  ->  ( ph  <->  ps ) )   =>    |-  ( A. x  x  =  y  ->  ( E. z ph  <->  E. z ps )
 )
 
Theoremdrnf2w2AUX7 29207* Weak version of drnf2 2027 not requiring ax-7 1745. (Contributed by NM, 25-Nov-2017.)
 |-  ( A. x  x  =  y  ->  ( ph  <->  ps ) )   =>    |-  ( A. x  x  =  y  ->  ( F/ z ph  <->  F/ z ps )
 )
 
Theoremdral2w3AUX7 29208 Weak version of dral2 2020 not requiring ax-7 1745. (Contributed by NM, 25-Nov-2017.)
 |-  ( A. x  x  =  y  ->  ( ph  <->  ps ) )   =>    |-  ( A. x  x  =  y  ->  (
 A. x ph  <->  A. x ps )
 )
 
Theoremdrex2w3AUX7 29209 Weak version of drex2 2025 not requiring ax-7 1745. (Contributed by NM, 25-Nov-2017.)
 |-  ( A. x  x  =  y  ->  ( ph  <->  ps ) )   =>    |-  ( A. x  x  =  y  ->  ( E. x ph  <->  E. x ps )
 )
 
Theoremdrnf2w3AUX7 29210 Weak version of drnf2 2027 not requiring ax-7 1745. (Contributed by NM, 25-Nov-2017.)
 |-  ( A. x  x  =  y  ->  ( ph  <->  ps ) )   =>    |-  ( A. x  x  =  y  ->  ( F/ x ph  <->  F/ x ps )
 )
 
TheoremexdistrfNEW7 29211 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.) (Revised by NM, 25-Nov-2017.) Revised to prove from ax-7v 29148 instead of ax-7 1745.
 |-  ( -.  A. x  x  =  y  ->  F/ y ph )   =>    |-  ( E. x E. y ( ph  /\  ps )  ->  E. x ( ph  /\ 
 E. y ps )
 )
 
Theoremdrsb1NEW7 29212 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 ) )
 
TheoremspimtNEW7 29213 Closed theorem form of spim 1955. (Contributed by NM, 15-Jan-2008.) (Revised by Mario Carneiro, 17-Oct-2016.)
 |-  (
 ( F/ x ps  /\ 
 A. x ( x  =  y  ->  ( ph  ->  ps ) ) ) 
 ->  ( A. x ph  ->  ps ) )
 
TheoremspimNEW7 29214 Specialization, using implicit substitution. Compare Lemma 14 of [Tarski] p. 70. The spim 1955 series of theorems requires that only one direction of the substitution hypothesis hold. (Contributed by NM, 5-Aug-1993.) (Revised by Mario Carneiro, 3-Oct-2016.)
 |-  F/ x ps   &    |-  ( x  =  y  ->  ( ph  ->  ps ) )   =>    |-  ( A. x ph 
 ->  ps )
 
TheoremspimeNEW7 29215 Existential introduction, using implicit substitution. Compare Lemma 14 of [Tarski] p. 70. (Contributed by NM, 7-Aug-1994.) (Revised by Mario Carneiro, 3-Oct-2016.)
 |-  F/ x ph   &    |-  ( x  =  y  ->  ( ph  ->  ps ) )   =>    |-  ( ph  ->  E. x ps )
 
TheoremspimedNEW7 29216 Deduction version of spime 1960. (Contributed by NM, 5-Aug-1993.) (Revised by Mario Carneiro, 3-Oct-2016.)
 |-  ( ch  ->  F/ x ph )   &    |-  ( x  =  y 
 ->  ( ph  ->  ps )
 )   =>    |-  ( ch  ->  ( ph  ->  E. x ps )
 )
 
Theoremcbv1hwAUX7 29217* Weak version of cbv1h 2031 not requiring ax-7 1745. (Contributed by NM, 28-Oct-2017.)
 |-  ( 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 ) )
 
Theoremcbv1wAUX7 29218* Weak version of cbv1 2032 not requiring ax-7 1745. (Contributed by NM, 28-Oct-2017.)
 |-  ( ph  ->  F/ y ps )   &    |-  ( ph  ->  F/ x ch )   &    |-  ( ph  ->  ( x  =  y  ->  ( ps  ->  ch ) ) )   =>    |-  ( A. x A. y ph  ->  ( A. x ps  ->  A. y ch )
 )
 
Theoremcbv2hwAUX7 29219* Weak version of cbv2h 2033 not requiring ax-7 1745. (Contributed by NM, 28-Oct-2017.)
 |-  ( 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 )
 )
 
Theoremcbv2wAUX7 29220* Weak version of cbv2 2034 not requiring ax-7 1745. (Contributed by NM, 28-Oct-2017.)
 |-  ( ph  ->  F/ y ps )   &    |-  ( ph  ->  F/ x ch )   &    |-  ( ph  ->  ( x  =  y  ->  ( ps  <->  ch ) ) )   =>    |-  ( A. x A. y ph  ->  ( A. x ps  <->  A. y ch )
 )
 
Theoremcbv3wAUX7 29221* Weak version of cbv3 2035 not requiring ax-7 1745. (Contributed by NM, 28-Oct-2017.)
 |-  F/ y ph   &    |-  F/ x ps   &    |-  ( x  =  y  ->  (
 ph  ->  ps ) )   =>    |-  ( A. x ph 
 ->  A. y ps )
 
Theoremcbv3hwAUX7 29222* Weak version of cbv3h 2036 not requiring ax-7 1745. (Contributed by NM, 28-Oct-2017.)
 |-  ( ph  ->  A. y ph )   &    |-  ( ps  ->  A. x ps )   &    |-  ( x  =  y  ->  (
 ph  ->  ps ) )   =>    |-  ( A. x ph 
 ->  A. y ps )
 
TheoremcbvalwwAUX7 29223* Weak version of cbval 2037 not requiring ax-7 1745. (Contributed by NM, 28-Oct-2017.)
 |-  F/ y ph   &    |-  F/ x ps   &    |-  ( x  =  y  ->  (
 ph 
 <->  ps ) )   =>    |-  ( A. x ph  <->  A. y ps )
 
TheoremcbvexwAUX7 29224* Weak version of cbvex 2038 not requiring ax-7 1745. (Contributed by NM, 28-Oct-2017.)
 |-  F/ y ph   &    |-  F/ x ps   &    |-  ( x  =  y  ->  (
 ph 
 <->  ps ) )   =>    |-  ( E. x ph  <->  E. y ps )
 
TheoremspimvNEW7 29225* A version of spim 1955 with a distinct variable requirement instead of a bound variable hypothesis. (Contributed by NM, 5-Aug-1993.)
 |-  ( x  =  y  ->  (
 ph  ->  ps ) )   =>    |-  ( A. x ph 
 ->  ps )
 
TheoremaevwAUX7 29226* Weak version of aev 2011 not requiring ax-7 1745. (Contributed by NM, 28-Oct-2017.)
 |-  ( A. x  x  =  y  ->  A. z  w  =  v )
 
TheoremaevNEW7 29227* 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 )
 
Theoremhbaew3AUX7 29228* Weak version of hbae 2005 not requiring ax-7 1745. Has different distinct variable requirements from hbaewAUX7 29184. (Contributed by NM, 30-Oct-2017.)
 |-  ( A. x  x  =  y  ->  A. z A. x  x  =  y )
 
Theoremnfaew3AUX7 29229* Weak version of nfae 2006 not requiring ax-7 1745. (Contributed by NM, 25-Nov-2017.)
 |-  F/ z A. x  x  =  y
 
Theoremnfnaew3AUX7 29230* Weak version of nfnae 2008 not requiring ax-7 1745. (Contributed by NM, 25-Nov-2017.)
 |-  F/ z  -.  A. x  x  =  y
 
TheoremequviniNEW7 29231 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.)
 |-  ( x  =  y  ->  E. z ( x  =  z  /\  z  =  y ) )
 
TheoremequveliNEW7 29232 A variable elimination law for equality with no distinct variable requirements. (Compare equviniNEW7 29231.) (Contributed by NM, 1-Mar-2013.) (Proof shortened by Mario Carneiro, 17-Oct-2016.) (Revised by NM, 25-Nov-2017.) Revised to prove from ax-7v 29148 instead of ax-7 1745.
 |-  ( A. z ( z  =  x  <->  z  =  y
 )  ->  x  =  y )
 
TheoremequvinNEW7 29233* 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 ) )
 
Theoremax11v2NEW7 29234* 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 ) ) ) )
 
Theoremax11a2NEW7 29235* 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 ) ) ) )
 
Theoremax11oNEW7 29236 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 )
 ) ) )
 
Theoremequs4NEW7 29237 Lemma used in proofs of substitution properties. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Mario Carneiro, 20-May-2014.)
 |-  ( A. x ( x  =  y  ->  ph )  ->  E. x ( x  =  y  /\  ph )
 )
 
Theoremequs5NEW7 29238 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 )
 ) )
 
Theoremequs5bAUX7 29239 Lemma used in proofs of substitution properties. (Contributed by NM, 27-Oct-2017.)
 |-  ( -.  A. x  x  =  y  ->  ( E. x ( x  =  y  /\  ph )  <->  A. x ( x  =  y  ->  ph ) ) )
 
Theoremax15NEW7 29240 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.)

 |-  ( -.  A. z  z  =  x  ->  ( -.  A. z  z  =  y 
 ->  ( x  e.  y  ->  A. z  x  e.  y ) ) )
 
Theoremsb2NEW7 29241 One direction of a simplified definition of substitution. (Contributed by NM, 5-Aug-1993.)
 |-  ( A. x ( x  =  y  ->  ph )  ->  [ y  /  x ] ph )
 
Theoremequsb1NEW7 29242 Substitution applied to an atomic wff. (Contributed by NM, 5-Aug-1993.)
 |-  [ y  /  x ] x  =  y
 
Theoremequsb2NEW7 29243 Substitution applied to an atomic wff. (Contributed by NM, 5-Aug-1993.)
 |-  [ y  /  x ] y  =  x
 
TheoremsbiedNEW7 29244 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 ) )
 
TheoremsbieNEW7 29245 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 )
 
Theoremhbsb2aNEW7 29246 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 )
 
Theoremhbsb2eNEW7 29247 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 )
 
Theoremhbsb3NEW7 29248 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 )
 
Theoremnfs1NEW7 29249 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
 
Theorema16gNEW7 29250* Generalization of ax16 2094. (Contributed by NM, 25-Jul-2015.)
 |-  ( A. x  x  =  y  ->  ( ph  ->  A. z ph ) )
 
Theorema16gbNEW7 29251* A generalization of axiom ax-16 2194. (Contributed by NM, 5-Aug-1993.)
 |-  ( A. x  x  =  y  ->  ( ph  <->  A. z ph )
 )
 
Theorema16nfwAUX7 29252* Weak version of a16nf 2100 not requiring ax-7 1745. (Contributed by NM, 10-Oct-2017.)
 |-  ( A. x  x  =  y  ->  F/ z ph )
 
Theoremax16NEW7 29253* 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 ) )
 
Theorema16nfNEW7 29254* 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.) (Revised by NM, 25-Nov-2017.) Revised to prove from ax-7v 29148 instead of ax-7 1745.
 |-  ( A. x  x  =  y  ->  F/ z ph )
 
Theoremax16iNEW7 29255* 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 ) )
 
Theoremsb4NEW7 29256 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 )
 ) )
 
Theoremsb4bNEW7 29257 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 )
 ) )
 
Theoremhbsb2NEW7 29258 Bound-variable hypothesis builder for substitution. (Contributed by NM, 5-Aug-1993.)
 |-  ( -.  A. x  x  =  y  ->  ( [
 y  /  x ] ph  ->  A. x [ y  /  x ] ph )
 )
 
Theoremstdpc4NEW7 29259 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 )
 
TheoremsbftNEW7 29260 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 ) )
 
TheoremsbfNEW7 29261 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 )
 
Theoremsbequ5wAUX7 29262* Weak version of sbequ5 2080 not requiring ax-7 1745. (Contributed by NM, 27-Oct-2017.)
 |-  ( [ w  /  z ] A. x  x  =  y  <->  A. x  x  =  y )
 
TheoremsbhNEW7 29263 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 )
 
Theoremsbf2NEW7 29264 Substitution has no effect on a bound variable. (Contributed by NM, 1-Jul-2005.)
 |-  ( [ y  /  x ] A. x ph  <->  A. x ph )
 
Theoremnfsb2NEW7 29265 Bound-variable hypothesis builder for substitution. (Contributed by Mario Carneiro, 4-Oct-2016.)
 |-  ( -.  A. x  x  =  y  ->  F/ x [ y  /  x ] ph )
 
TheoremsbnNEW7 29266 Negation inside and outside of substitution are equivalent. (Contributed by NM, 5-Aug-1993.)
 |-  ( [ y  /  x ]  -.  ph  <->  -.  [ y  /  x ] ph )
 
Theoremsbi1NEW7 29267 Removal of implication from substitution. (Contributed by NM, 5-Aug-1993.)
 |-  ( [ y  /  x ] ( ph  ->  ps )  ->  ( [
 y  /  x ] ph  ->  [ y  /  x ] ps ) )
 
Theoremsbi2NEW7 29268 Introduction of implication into substitution. (Contributed by NM, 5-Aug-1993.)
 |-  (
 ( [ y  /  x ] ph  ->  [ y  /  x ] ps )  ->  [ y  /  x ] ( ph  ->  ps ) )
 
TheoremsbimNEW7 29269 Implication inside and outside of substitution are equivalent. (Contributed by NM, 5-Aug-1993.)
 |-  ( [ y  /  x ] ( ph  ->  ps )  <->  ( [ y  /  x ] ph  ->  [ y  /  x ] ps ) )
 
TheoremsborNEW7 29270 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 ) )
 
TheoremsbrimNEW7 29271 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 ) )
 
TheoremsblimNEW7 29272 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 ) )
 
TheoremsbanNEW7 29273 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 ) )
 
TheoremsbbiNEW7 29274 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 ) )
 
TheoremspsbeNEW7 29275 A specialization theorem. (Contributed by NM, 5-Aug-1993.)
 |-  ( [ y  /  x ] ph  ->  E. x ph )
 
TheoremspsbimNEW7 29276 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 ) )
 
TheoremspsbbiNEW7 29277 Specialization of biconditional. (Contributed by NM, 5-Aug-1993.)
 |-  ( A. x ( ph  <->  ps )  ->  ( [ y  /  x ] ph  <->  [ y  /  x ] ps ) )
 
TheoremsbbidNEW7 29278 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 ) )
 
Theoremsbequ8NEW7 29279 Elimination of equality from antecedent after substitution. (Contributed by NM, 5-Aug-1993.)
 |-  ( [ y  /  x ] ph  <->  [ y  /  x ] ( x  =  y  ->  ph ) )
 
Theoremnfsb4twAUX7 29280* Weak version of nfsb4t 2129 not requiring ax-7 1745. (Contributed by NM, 27-Oct-2017.)
 |-  ( A. x F/ z ph  ->  ( -.  A. z  z  =  y  ->  F/ z [ y  /  x ] ph ) )
 
Theoremnfsb4wAUX7 29281* Weak version of nfsb4t 2129 not requiring ax-7 1745. (Contributed by NM, 27-Oct-2017.)
 |-  F/ z ph   =>    |-  ( -.  A. z  z  =  y  ->  F/ z [ y  /  x ] ph )
 
TheoremsbequiNEW7 29282 An equality theorem for substitution. (Contributed by NM, 5-Aug-1993.)
 |-  ( x  =  y  ->  ( [ x  /  z ] ph  ->  [ y  /  z ] ph )
 )
 
TheoremsbequNEW7 29283 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 ) )
 
Theoremdrsb2NEW7 29284 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 ) )
 
TheoremsbcoNEW7 29285 A composition law for substitution. (Contributed by NM, 5-Aug-1993.)
 |-  ( [ y  /  x ] [ x  /  y ] ph  <->  [ y  /  x ] ph )
 
Theoremsbid2NEW7 29286 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 )
 
TheoremsbidmNEW7 29287 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 )
 
Theoremsbco2wAUX7 29288* Weak version of sbco2 2135 not requiring ax-7 1745. (Contributed by NM, 27-Oct-2017.)
 |-  F/ z ph   =>    |-  ( [ y  /  z ] [ z  /  x ] ph  <->  [ y  /  x ] ph )
 
Theoremsbco2dwAUX7 29289* Weak version of sbco2d 2136 not requiring ax-7 1745. (Contributed by NM, 27-Oct-2017.)
 |-  F/ x ph   &    |-  F/ z ph   &    |-  ( ph  ->  F/ z ps )   =>    |-  ( ph  ->  ( [ y  /  z ] [ z  /  x ] ps  <->  [ y  /  x ] ps ) )
 
Theoremsbco3wAUX7 29290* Weak version of sbco3 2137 not requiring ax-7 1745. (Contributed by NM, 27-Oct-2017.)
 |-  ( [ z  /  y ] [ y  /  x ] ph  <->  [ z  /  x ] [ x  /  y ] ph )
 
TheoremsbcomwAUX7 29291* Weak version of sbcom 2138 not requiring ax-7 1745. (Contributed by NM, 27-Oct-2017.)
 |-  ( [ y  /  z ] [ y  /  x ] ph  <->  [ y  /  x ] [ y  /  z ] ph )
 
Theoremsb5rfNEW7 29292 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 ) )
 
Theoremsb6rfNEW7 29293 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 )
 )
 
Theoremsb8wAUX7 29294* Weak version of sb8 2141 not requiring ax-7 1745. (Contributed by NM, 28-Oct-2017.)
 |-  F/ y ph   =>    |-  ( A. x ph  <->  A. y [ y  /  x ] ph )
 
Theoremsb8ewAUX7 29295* Weak version of sb8e 2142 not requiring ax-7 1745. (Contributed by NM, 28-Oct-2017.)
 |-  F/ y ph   =>    |-  ( E. x ph  <->  E. y [ y  /  x ] ph )
 
Theoremax11vNEW7 29296* This is a version of ax-11o 2191 when the variables are distinct. Axiom (C8) of [Monk2] p. 105. See theorem ax11v2 2045 for the rederivation of ax-11o 2191 from this theorem. (Contributed by NM, 5-Aug-1993.)
 |-  ( x  =  y  ->  (
 ph  ->  A. x ( x  =  y  ->  ph )
 ) )
 
Theoremsb56NEW7 29297* 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 1656. (Contributed by NM, 14-Apr-2008.)
 |-  ( E. x ( x  =  y  /\  ph )  <->  A. x ( x  =  y  ->  ph ) )
 
Theoremsb6NEW7 29298* 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 )
 )
 
Theoremsb5NEW7 29299* 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 )
 )
 
TheoremexsbNEW7 29300* An equivalent expression for existence. (Contributed by NM, 2-Feb-2005.) (Revised by NM, 28-Nov-2017.) Revised to prove from ax-7v 29148 instead of ax-7 1745.
 |-  ( E. x ph  <->  E. y A. x ( x  =  y  -> 
 ph ) )
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