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Theorem List for Metamath Proof Explorer - 29501-29600   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremnfnaew3AUX7 29501* Weak version of nfnae 1909 not requiring ax-7 1720. (Contributed by NM, 25-Nov-2017.)
 |-  F/ z  -.  A. x  x  =  y
 
TheoremequviniNEW7 29502 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 29503 A variable elimination law for equality with no distinct variable requirements. (Compare equviniNEW7 29502.) (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 29419 instead of ax-7 1720.
 |-  ( A. z ( z  =  x  <->  z  =  y
 )  ->  x  =  y )
 
TheoremequvinNEW7 29504* 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 29505* Recovery of ax-11o 2093 from ax11v 2049. This proof uses ax-10 2092 and ax-11 1727. 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 29506* Derive ax-11o 2093 from a hypothesis in the form of ax-11 1727. ax-10 2092 and ax-11 1727 are used by the proof, but not ax-10o 2091 or ax-11o 2093. 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 29507 Derivation of set.mm's original ax-11o 2093 from ax-10 2092 and the shorter ax-11 1727 that has replaced it.

An open problem is whether this theorem can be proved without relying on ax-16 2096 or ax-17 1606 (given all of the original and new versions of sp 1728 through ax-15 2095).

Another open problem is whether this theorem can be proved without relying on ax12o 1887.

Theorem ax11 2107 shows the reverse derivation of ax-11 1727 from ax-11o 2093.

Normally, ax11o 1947 should be used rather than ax-11o 2093, 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 29508 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 29509 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 29510 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 29511 Axiom ax-15 2095 is redundant if we assume ax-17 1606. 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 1973 and ax-17 1606. 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 29512 One direction of a simplified definition of substitution. (Contributed by NM, 5-Aug-1993.)
 |-  ( A. x ( x  =  y  ->  ph )  ->  [ y  /  x ] ph )
 
Theoremequsb1NEW7 29513 Substitution applied to an atomic wff. (Contributed by NM, 5-Aug-1993.)
 |-  [ y  /  x ] x  =  y
 
Theoremequsb2NEW7 29514 Substitution applied to an atomic wff. (Contributed by NM, 5-Aug-1993.)
 |-  [ y  /  x ] y  =  x
 
TheoremsbiedNEW7 29515 Conversion of implicit substitution to explicit substitution (deduction version of sbie 1991). (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 29516 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 29517 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 29518 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 29519 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 29520 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 29521* Generalization of ax16 1998. (Contributed by NM, 25-Jul-2015.)
 |-  ( A. x  x  =  y  ->  ( ph  ->  A. z ph ) )
 
Theorema16gbNEW7 29522* A generalization of axiom ax-16 2096. (Contributed by NM, 5-Aug-1993.)
 |-  ( A. x  x  =  y  ->  ( ph  <->  A. z ph )
 )
 
Theorema16nfwAUX7 29523* Weak version of a16nf 2004 not requiring ax-7 1720. (Contributed by NM, 10-Oct-2017.)
 |-  ( A. x  x  =  y  ->  F/ z ph )
 
Theoremax16NEW7 29524* Proof of older axiom ax-16 2096. (Contributed by NM, 8-Nov-2006.) (Revised by NM, 22-Sep-2017.)
 |-  ( A. x  x  =  y  ->  ( ph  ->  A. x ph ) )
 
Theorema16nfNEW7 29525* If dtru 4217 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 29419 instead of ax-7 1720.
 |-  ( A. x  x  =  y  ->  F/ z ph )
 
Theoremax16iNEW7 29526* Inference with ax16 1998 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 29527 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 29528 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 29529 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 29530 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 3016 and rspsbc 3082. (Contributed by NM, 5-Aug-1993.)
 |-  ( A. x ph  ->  [ y  /  x ] ph )
 
TheoremsbftNEW7 29531 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 29532 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 29533* Weak version of sbequ5 1984 not requiring ax-7 1720. (Contributed by NM, 27-Oct-2017.)
 |-  ( [ w  /  z ] A. x  x  =  y  <->  A. x  x  =  y )
 
TheoremsbhNEW7 29534 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 29535 Substitution has no effect on a bound variable. (Contributed by NM, 1-Jul-2005.)
 |-  ( [ y  /  x ] A. x ph  <->  A. x ph )
 
Theoremnfsb2NEW7 29536 Bound-variable hypothesis builder for substitution. (Contributed by Mario Carneiro, 4-Oct-2016.)
 |-  ( -.  A. x  x  =  y  ->  F/ x [ y  /  x ] ph )
 
TheoremsbnNEW7 29537 Negation inside and outside of substitution are equivalent. (Contributed by NM, 5-Aug-1993.)
 |-  ( [ y  /  x ]  -.  ph  <->  -.  [ y  /  x ] ph )
 
Theoremsbi1NEW7 29538 Removal of implication from substitution. (Contributed by NM, 5-Aug-1993.)
 |-  ( [ y  /  x ] ( ph  ->  ps )  ->  ( [
 y  /  x ] ph  ->  [ y  /  x ] ps ) )
 
Theoremsbi2NEW7 29539 Introduction of implication into substitution. (Contributed by NM, 5-Aug-1993.)
 |-  (
 ( [ y  /  x ] ph  ->  [ y  /  x ] ps )  ->  [ y  /  x ] ( ph  ->  ps ) )
 
TheoremsbimNEW7 29540 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 29541 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 29542 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 29543 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 29544 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 29545 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 29546 A specialization theorem. (Contributed by NM, 5-Aug-1993.)
 |-  ( [ y  /  x ] ph  ->  E. x ph )
 
TheoremspsbimNEW7 29547 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 29548 Specialization of biconditional. (Contributed by NM, 5-Aug-1993.)
 |-  ( A. x ( ph  <->  ps )  ->  ( [ y  /  x ] ph  <->  [ y  /  x ] ps ) )
 
TheoremsbbidNEW7 29549 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 29550 Elimination of equality from antecedent after substitution. (Contributed by NM, 5-Aug-1993.)
 |-  ( [ y  /  x ] ph  <->  [ y  /  x ] ( x  =  y  ->  ph ) )
 
Theoremnfsb4twAUX7 29551* Weak version of nfsb4t 2033 not requiring ax-7 1720. (Contributed by NM, 27-Oct-2017.)
 |-  ( A. x F/ z ph  ->  ( -.  A. z  z  =  y  ->  F/ z [ y  /  x ] ph ) )
 
Theoremnfsb4wAUX7 29552* Weak version of nfsb4t 2033 not requiring ax-7 1720. (Contributed by NM, 27-Oct-2017.)
 |-  F/ z ph   =>    |-  ( -.  A. z  z  =  y  ->  F/ z [ y  /  x ] ph )
 
TheoremsbequiNEW7 29553 An equality theorem for substitution. (Contributed by NM, 5-Aug-1993.)
 |-  ( x  =  y  ->  ( [ x  /  z ] ph  ->  [ y  /  z ] ph )
 )
 
TheoremsbequNEW7 29554 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 29555 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 29556 A composition law for substitution. (Contributed by NM, 5-Aug-1993.)
 |-  ( [ y  /  x ] [ x  /  y ] ph  <->  [ y  /  x ] ph )
 
Theoremsbid2NEW7 29557 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 29558 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 29559* Weak version of sbco2 2039 not requiring ax-7 1720. (Contributed by NM, 27-Oct-2017.)
 |-  F/ z ph   =>    |-  ( [ y  /  z ] [ z  /  x ] ph  <->  [ y  /  x ] ph )
 
Theoremsbco2dwAUX7 29560* Weak version of sbco2d 2040 not requiring ax-7 1720. (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 29561* Weak version of sbco3 2041 not requiring ax-7 1720. (Contributed by NM, 27-Oct-2017.)
 |-  ( [ z  /  y ] [ y  /  x ] ph  <->  [ z  /  x ] [ x  /  y ] ph )
 
TheoremsbcomwAUX7 29562* Weak version of sbcom 2042 not requiring ax-7 1720. (Contributed by NM, 27-Oct-2017.)
 |-  ( [ y  /  z ] [ y  /  x ] ph  <->  [ y  /  x ] [ y  /  z ] ph )
 
Theoremsb5rfNEW7 29563 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 29564 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 29565* Weak version of sb8 2045 not requiring ax-7 1720. (Contributed by NM, 28-Oct-2017.)
 |-  F/ y ph   =>    |-  ( A. x ph  <->  A. y [ y  /  x ] ph )
 
Theoremsb8ewAUX7 29566* Weak version of sb8e 2046 not requiring ax-7 1720. (Contributed by NM, 28-Oct-2017.)
 |-  F/ y ph   =>    |-  ( E. x ph  <->  E. y [ y  /  x ] ph )
 
Theoremax11vNEW7 29567* This is a version of ax-11o 2093 when the variables are distinct. Axiom (C8) of [Monk2] p. 105. See theorem ax11v2 1945 for the rederivation of ax-11o 2093 from this theorem. (Contributed by NM, 5-Aug-1993.)
 |-  ( x  =  y  ->  (
 ph  ->  A. x ( x  =  y  ->  ph )
 ) )
 
Theoremsb56NEW7 29568* 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 1639. (Contributed by NM, 14-Apr-2008.)
 |-  ( E. x ( x  =  y  /\  ph )  <->  A. x ( x  =  y  ->  ph ) )
 
Theoremsb6NEW7 29569* 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 29570* 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 29571* An equivalent expression for existence. (Contributed by NM, 2-Feb-2005.) (Revised by NM, 28-Nov-2017.) Revised to prove from ax-7v 29419 instead of ax-7 1720.
 |-  ( E. x ph  <->  E. y A. x ( x  =  y  -> 
 ph ) )
 
TheoremexsbOLDNEW7 29572* An equivalent expression for existence. Obsolete as of 19-Jun-2017. (Contributed by NM, 2-Feb-2005.) (New usage is discouraged.)
 |-  ( E. x ph  <->  E. y A. x ( x  =  y  -> 
 ph ) )
 
Theoremequsb3lemNEW7 29573* Lemma for equsb3 2054. (Contributed by Raph Levien and FL, 4-Dec-2005.) (Proof shortened by Andrew Salmon, 14-Jun-2011.)
 |-  ( [ x  /  y ] y  =  z  <->  x  =  z )
 
Theoremequsb3NEW7 29574* Substitution applied to an atomic wff. (Contributed by Raph Levien and FL, 4-Dec-2005.)
 |-  ( [ x  /  y ] y  =  z  <->  x  =  z )
 
Theoremelsb3NEW7 29575* 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 )
 
Theoremelsb4NEW7 29576* 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 )
 
Theoremhbs1NEW7 29577*  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 )
 
Theoremnfs1vNEW7 29578*  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
 
TheoremsbhbwAUX7 29579* Weak version of sbhb 2059 not requiring ax-7 1720. (Contributed by NM, 28-Oct-2017.)
 |-  (
 ( ph  ->  A. x ph )  <->  A. y ( ph  ->  [ y  /  x ] ph ) )
 
Theoremsbnf2NEW7 29580* 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 ) )
 
Theorem2sb5NEW7 29581* Equivalence for double substitution. (Contributed by NM, 3-Feb-2005.)
 |-  ( [ z  /  x ] [ w  /  y ] ph  <->  E. x E. y
 ( ( x  =  z  /\  y  =  w )  /\  ph )
 )
 
Theorem2sb6NEW7 29582* Equivalence for double substitution. (Contributed by NM, 3-Feb-2005.)
 |-  ( [ z  /  x ] [ w  /  y ] ph  <->  A. x A. y
 ( ( x  =  z  /\  y  =  w )  ->  ph )
 )
 
Theoremsb6aNEW7 29583* Equivalence for substitution. (Contributed by NM, 5-Aug-1993.)
 |-  ( [ y  /  x ] ph  <->  A. x ( x  =  y  ->  [ x  /  y ] ph )
 )
 
Theoremsbid2vNEW7 29584* 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 )
 
TheoremsbelxNEW7 29585* Elimination of substitution. (Contributed by NM, 5-Aug-1993.)
 |-  ( ph 
 <-> 
 E. x ( x  =  y  /\  [ x  /  y ] ph ) )
 
Theoremsbel2xNEW7 29586* Elimination of double substitution. (Contributed by NM, 5-Aug-1993.)
 |-  ( ph 
 <-> 
 E. x E. y
 ( ( x  =  z  /\  y  =  w )  /\  [
 y  /  w ] [ x  /  z ] ph ) )
 
Theoremsbal1NEW7 29587* A theorem used in elimination of disjoint variable restriction on  x and  y by replacing it with a distinctor  -.  A. x x  =  z. (Contributed by NM, 5-Aug-1993.)
 |-  ( -.  A. x  x  =  z  ->  ( [
 z  /  y ] A. x ph  <->  A. x [ z  /  y ] ph )
 )
 
TheoremsbalNEW7 29588* Move universal quantifier in and out of substitution. (Contributed by NM, 5-Aug-1993.)
 |-  ( [ z  /  y ] A. x ph  <->  A. x [ z  /  y ] ph )
 
TheoremsbexNEW7 29589* Move existential quantifier in and out of substitution. (Contributed by NM, 27-Sep-2003.)
 |-  ( [ z  /  y ] E. x ph  <->  E. x [ z  /  y ] ph )
 
TheoremsbalvNEW7 29590* Quantify with new variable inside substitution. (Contributed by NM, 18-Aug-1993.)
 |-  ( [ y  /  x ] ph  <->  ps )   =>    |-  ( [ y  /  x ] A. z ph  <->  A. z ps )
 
TheoremnaecomsNEW7 29591 A commutation rule for distinct variable specifiers. (Contributed by NM, 2-Jan-2002.)
 |-  ( -.  A. x  x  =  y  ->  ph )   =>    |-  ( -.  A. y  y  =  x  -> 
 ph )
 
TheoremchvarNEW7 29592 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
 
Theoremequs45fNEW7 29593 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 )
 )
 
Theoremax11bNEW7 29594 A bidirectional version of ax11oNEW7 29507. (Contributed by NM, 30-Jun-2006.)
 |-  (
 ( -.  A. x  x  =  y  /\  x  =  y )  ->  ( ph  <->  A. x ( x  =  y  ->  ph )
 ) )
 
TheoremspvNEW7 29595* Specialization, using implicit substitution. (Contributed by NM, 30-Aug-1993.)
 |-  ( x  =  y  ->  (
 ph 
 <->  ps ) )   =>    |-  ( A. x ph 
 ->  ps )
 
TheoremspimevNEW7 29596* Distinct-variable version of spimeNEW7 29486. (Contributed by NM, 5-Aug-1993.)
 |-  ( x  =  y  ->  (
 ph  ->  ps ) )   =>    |-  ( ph  ->  E. x ps )
 
TheoremspeivNEW7 29597* Inference from existential specialization, using implicit substitution. (Contributed by NM, 19-Aug-1993.)
 |-  ( x  =  y  ->  (
 ph 
 <->  ps ) )   &    |-  ps   =>    |-  E. x ph
 
TheoremchvarvNEW7 29598* Implicit substitution of  y for  x into a theorem. (Contributed by NM, 20-Apr-1994.)
 |-  ( x  =  y  ->  (
 ph 
 <->  ps ) )   &    |-  ph   =>    |- 
 ps
 
TheoremcleljustNEW7 29599* When the class variables in definition df-clel 2292 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 1697 with the class variables in wcel 1696. 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 ) )
 
TheoremcleljustALTNEW7 29600* When the class variables in definition df-clel 2292 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 1697 with the class variables in wcel 1696. (Contributed by NM, 28-Jan-2004.) (Revised by Mario Carneiro, 21-Dec-2016.)
 |-  ( x  e.  y  <->  E. z ( z  =  x  /\  z  e.  y ) )
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