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Theorem List for Metamath Proof Explorer - 29301-29400   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
TheoremexsbOLDNEW7 29301* 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 29302* Lemma for equsb3 2151. (Contributed by Raph Levien and FL, 4-Dec-2005.) (Proof shortened by Andrew Salmon, 14-Jun-2011.)
 |-  ( [ x  /  y ] y  =  z  <->  x  =  z )
 
Theoremequsb3NEW7 29303* Substitution applied to an atomic wff. (Contributed by Raph Levien and FL, 4-Dec-2005.)
 |-  ( [ x  /  y ] y  =  z  <->  x  =  z )
 
Theoremelsb3NEW7 29304* 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 29305* 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 29306*  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 29307*  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 29308* Weak version of sbhb 2156 not requiring ax-7 1745. (Contributed by NM, 28-Oct-2017.)
 |-  (
 ( ph  ->  A. x ph )  <->  A. y ( ph  ->  [ y  /  x ] ph ) )
 
Theoremsbnf2NEW7 29309* 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 29310* 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 29311* 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 29312* Equivalence for substitution. (Contributed by NM, 5-Aug-1993.)
 |-  ( [ y  /  x ] ph  <->  A. x ( x  =  y  ->  [ x  /  y ] ph )
 )
 
Theoremsbid2vNEW7 29313* 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 29314* Elimination of substitution. (Contributed by NM, 5-Aug-1993.)
 |-  ( ph 
 <-> 
 E. x ( x  =  y  /\  [ x  /  y ] ph ) )
 
Theoremsbel2xNEW7 29315* 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 29316* 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 29317* 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 29318* 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 29319* 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 29320 A commutation rule for distinct variable specifiers. (Contributed by NM, 2-Jan-2002.)
 |-  ( -.  A. x  x  =  y  ->  ph )   =>    |-  ( -.  A. y  y  =  x  -> 
 ph )
 
TheoremchvarNEW7 29321 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 29322 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 29323 A bidirectional version of ax11oNEW7 29236. (Contributed by NM, 30-Jun-2006.)
 |-  (
 ( -.  A. x  x  =  y  /\  x  =  y )  ->  ( ph  <->  A. x ( x  =  y  ->  ph )
 ) )
 
TheoremspvNEW7 29324* Specialization, using implicit substitution. (Contributed by NM, 30-Aug-1993.)
 |-  ( x  =  y  ->  (
 ph 
 <->  ps ) )   =>    |-  ( A. x ph 
 ->  ps )
 
TheoremspimevNEW7 29325* Distinct-variable version of spimeNEW7 29215. (Contributed by NM, 5-Aug-1993.)
 |-  ( x  =  y  ->  (
 ph  ->  ps ) )   =>    |-  ( ph  ->  E. x ps )
 
TheoremspeivNEW7 29326* Inference from existential specialization, using implicit substitution. (Contributed by NM, 19-Aug-1993.)
 |-  ( x  =  y  ->  (
 ph 
 <->  ps ) )   &    |-  ps   =>    |-  E. x ph
 
TheoremchvarvNEW7 29327* Implicit substitution of  y for  x into a theorem. (Contributed by NM, 20-Apr-1994.)
 |-  ( x  =  y  ->  (
 ph 
 <->  ps ) )   &    |-  ph   =>    |- 
 ps
 
TheoremcleljustNEW7 29328* When the class variables in definition df-clel 2400 are replaced with set variables, this theorem of predicate calculus is the result. This theorem provides part of the justification for the consistency of that definition, which "overloads" the set variables in wel 1722 with the class variables in wcel 1721. Note: This proof is referenced on the Metamath Proof Explorer Home Page and shouldn't be changed. (Contributed by NM, 28-Jan-2004.) (Proof modification is discouraged.)
 |-  ( x  e.  y  <->  E. z ( z  =  x  /\  z  e.  y ) )
 
TheoremcleljustALTNEW7 29329* When the class variables in definition df-clel 2400 are replaced with set variables, this theorem of predicate calculus is the result. This theorem provides part of the justification for the consistency of that definition, which "overloads" the set variables in wel 1722 with the class variables in wcel 1721. (Contributed by NM, 28-Jan-2004.) (Revised by Mario Carneiro, 21-Dec-2016.)
 |-  ( x  e.  y  <->  E. z ( z  =  x  /\  z  e.  y ) )
 
Theoremsb6xNEW7 29330 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 )
 )
 
Theoremnfs1fNEW7 29331 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
 
TheoremsbtNEW7 29332 A substitution into a theorem remains true. (See chvarNEW7 29321 and chvarvNEW7 29327 for versions using implicit substitution.) (Contributed by NM, 21-Jan-2004.) (Proof shortened by Andrew Salmon, 25-May-2011.)
 |-  ph   =>    |- 
 [ y  /  x ] ph
 
TheoremsbiedvNEW7 29333* Conversion of implicit substitution to explicit substitution (deduction version of sbieNEW7 29245). (Contributed by NM, 7-Jan-2017.)
 |-  (
 ( ph  /\  x  =  y )  ->  ( ps 
 <->  ch ) )   =>    |-  ( ph  ->  ( [ y  /  x ] ps  <->  ch ) )
 
Theoremsb6fNEW7 29334 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 )
 )
 
Theoremsb5fNEW7 29335 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 )
 )
 
Theoremax16ALTNEW7 29336* Alternate proof of ax16NEW7 29253. (Contributed by NM, 17-May-2008.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  ( A. x  x  =  y  ->  ( ph  ->  A. x ph ) )
 
TheoremdvelimALTNEW7 29337* Version of dvelim 2066 that doesn't use ax-10 2190. (See dvelimh 2015 for a version that doesn't use ax-11 1757.) (Contributed by NM, 17-May-2008.) (New usage is discouraged.) (Proof modification is discouraged.)
 |-  ( ph  ->  A. x ph )   &    |-  (
 z  =  y  ->  ( ph  <->  ps ) )   =>    |-  ( -.  A. x  x  =  y  ->  ( ps  ->  A. x ps ) )
 
Theoremsb3NEW7 29338 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 ) )
 
Theoremdfsb2NEW7 29339 An alternate definition of proper substitution that, like df-sb 1656, 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 )
 ) )
 
Theoremdfsb3NEW7 29340 An alternate definition of proper substitution df-sb 1656 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 )
 ) )
 
Theoremsb3anNEW7 29341 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 ) )
 
TheoremsblbisNEW7 29342 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 ) )
 
TheoremsbrbisNEW7 29343 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 ) )
 
TheoremsbrbifNEW7 29344 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 ) )
 
Theoremax7w1AUX7 29345 Weak version of ax-7 1745 not requiring ax-7 1745. (Contributed by NM, 9-Oct-2017.)
 |-  ( A. x  x  =  y  ->  ( A. x A. y ph  ->  A. y A. x ph ) )
 
Theoremax7w1hAUX7 29346 Weak version of hbal 1747 not requiring ax-7 1745. (Contributed by NM, 27-Oct-2017.)
 |-  ( A. x  x  =  y  ->  ( ph  ->  A. x ph ) )   =>    |-  ( A. x  x  =  y  ->  ( A. y ph  ->  A. x A. y ph ) )
 
Theoremhbaew0AUX7 29347 Weak version of hbae 2005 not requiring ax-7 1745. (Contributed by NM, 29-Oct-2017.)
 |-  ( A. x  x  =  y  ->  A. z  x  =  y )
 
Theoremhbaew4AUX7 29348 Weak version of hbae 2005 not requiring ax-7 1745. (Contributed by NM, 30-Oct-2017.)
 |-  ( A. x  x  =  z  ->  ( A. x  x  =  y  ->  A. z A. x  x  =  y ) )
 
Theoremhbaew5AUX7 29349* Weak version of hbae 2005 not requiring ax-7 1745. (Contributed by NM, 30-Oct-2017.)
 |-  ( A. u  u  =  v  ->  ( A. x  x  =  y  ->  A. z A. x  x  =  y ) )
 
Theoremax7w2AUX7 29350 Special case of ax-7 1745. (Contributed by NM, 9-Oct-2017.)
 |-  ( A. x A. y [
 y  /  x ] ph  ->  A. y A. x [ y  /  x ] ph )
 
Theoremax7w3AUX7 29351 Special case of ax-7 1745. (Contributed by NM, 12-Oct-2017.)
 |-  ( A. x A. y  x  =  y  ->  A. y A. x  x  =  y )
 
Theoremax7w6AUX7 29352 Version of sb9i 2143 with the usage of ax-7 1745 broken out as a hypothesis. (Contributed by NM, 16-Oct-2017.)
 |-  ( A. x A. y [ x  /  y ] ph  ->  A. y A. x [ x  /  y ] ph )   =>    |-  ( A. x [ x  /  y ] ph  ->  A. y [ y  /  x ] ph )
 
Theoremax7w7AUX7 29353 Special case of ax-7 1745. (Contributed by NM, 12-Oct-2017.)
 |-  ( A. x A. y  -.  x  =  y  ->  A. y A. x  -.  x  =  y )
 
Theoremax7w8AUX7 29354 Special case of ax-7 1745. (Contributed by NM, 13-Oct-2017.)
 |-  ( ph  ->  A. x ph )   =>    |-  ( A. x A. y ph  ->  A. y A. x ph )
 
Theoremax7nfAUX7 29355 Special case of ax-7 1745. (Contributed by NM, 23-Nov-2017.)
 |-  F/ x ph   =>    |-  ( A. x A. y ph  ->  A. y A. x ph )
 
Theoremax7w7tAUX7 29356 Special case of ax-7 1745. (Contributed by NM, 23-Nov-2017.)
 |-  ( A. y ( ph  ->  A. x ph )  ->  ( A. x A. y ph  ->  A. y A. x ph ) )
 
Theoremax7wnftAUX7 29357 Special case of ax-7 1745. (Contributed by NM, 23-Nov-2017.)
 |-  ( A. y F/ x ph  ->  ( A. x A. y ph  ->  A. y A. x ph ) )
 
Theoremax7w4AUX7 29358 Remove quantifier from ax-7 1745. (Contributed by NM, 12-Oct-2017.)
 |-  ( A. x A. y A. y ph  ->  A. y A. x A. y ph )   =>    |-  ( A. x A. y ph  ->  A. y A. x ph )
 
Theoremax7w5AUX7 29359 Same as hbal 1747 with explicit ax-7 1745 hypothesis. (Contributed by NM, 12-Oct-2017.)
 |-  ( ph  ->  A. x ph )   &    |-  ( A. y A. x ph  ->  A. x A. y ph )   =>    |-  ( A. y ph  ->  A. x A. y ph )
 
Theoremax7w9AUX7 29360 Special case of ax-7 1745 proved from ax-7v 29148. (Contributed by NM, 28-Nov-2017.)
 |-  ( A. x A. y ( x  =  y  /\  x  =  z )  ->  A. y A. x ( x  =  y  /\  x  =  z
 ) )
 
Theoremalcomw9bAUX7 29361 Special case of alcom 1748 proved from ax-7v 29148. (Contributed by NM, 28-Nov-2017.)
 |-  ( A. x A. y ( x  =  y  /\  x  =  z )  <->  A. y A. x ( x  =  y  /\  x  =  z )
 )
 
19.26.1.2  Theorems derived from ax-7 (suffix OLD7)
 
Axiomax-7OLD7 29362 Axiom of Quantifier Commutation. This axiom says universal quantifiers can be swapped. One of the 4 axioms of pure predicate calculus. Axiom scheme C6' in [Megill] p. 448 (p. 16 of the preprint). Also appears as Lemma 12 of [Monk2] p. 109 and Axiom C5-3 of [Monk2] p. 113. This axiom scheme is logically redundant (see ax7w 1729) but is used as an auxiliary axiom to achieve metalogical completeness. (Contributed by NM, 5-Aug-1993.)
 |-  ( A. x A. y ph  ->  A. y A. x ph )
 
Theorema7sOLD7 29363 Swap quantifiers in an antecedent. (Contributed by NM, 5-Aug-1993.)
 |-  ( A. x A. y ph  ->  ps )   =>    |-  ( A. y A. x ph  ->  ps )
 
TheoremhbalOLD7 29364 If  x is not free in  ph, it is not free in  A. y ph. (Contributed by NM, 5-Aug-1993.)
 |-  ( ph  ->  A. x ph )   =>    |-  ( A. y ph  ->  A. x A. y ph )
 
TheoremalcomOLD7 29365 Theorem 19.5 of [Margaris] p. 89. (Contributed by NM, 5-Aug-1993.)
 |-  ( A. x A. y ph  <->  A. y A. x ph )
 
Theoremalrot3OLD7 29366 Theorem *11.21 in [WhiteheadRussell] p. 160. (Contributed by Andrew Salmon, 24-May-2011.)
 |-  ( A. x A. y A. z ph  <->  A. y A. z A. x ph )
 
Theoremalrot4OLD7 29367 Rotate 4 universal quantifiers twice. (Contributed by NM, 2-Feb-2005.) (Proof shortened by Fan Zheng, 6-Jun-2016.)
 |-  ( A. x A. y A. z A. w ph  <->  A. z A. w A. x A. y ph )
 
TheoremhbaldOLD7 29368 Deduction form of bound-variable hypothesis builder hbalOLD7 29364. (Contributed by NM, 2-Jan-2002.)
 |-  ( ph  ->  A. y ph )   &    |-  ( ph  ->  ( ps  ->  A. x ps ) )   =>    |-  ( ph  ->  ( A. y ps  ->  A. x A. y ps ) )
 
TheoremhbexOLD7 29369 If  x is not free in  ph, it is not free in  E. y ph. (Contributed by NM, 5-Aug-1993.)
 |-  ( ph  ->  A. x ph )   =>    |-  ( E. y ph  ->  A. x E. y ph )
 
Theorem19.12OLD7 29370 Theorem 19.12OLD7 of [Margaris] p. 89. Assuming the converse is a mistake sometimes made by beginners! But sometimes the converse does hold, as in 19.12vvOLD7 29385 and r19.12sn 3832. (Contributed by NM, 5-Aug-1993.)
 |-  ( E. x A. y ph  ->  A. y E. x ph )
 
TheoremnfalOLD7 29371 If  x is not free in  ph, it is not free in  A. y ph. (Contributed by Mario Carneiro, 11-Aug-2016.)
 |-  F/ x ph   =>    |- 
 F/ x A. y ph
 
TheoremnfexOLD7 29372 If  x is not free in  ph, it is not free in  E. y ph. (Contributed by Mario Carneiro, 11-Aug-2016.)
 |-  F/ x ph   =>    |- 
 F/ x E. y ph
 
TheoremnfnfOLD7 29373 If  x is not free in  ph, it is not free in  F/ y ph. (Contributed by Mario Carneiro, 11-Aug-2016.)
 |-  F/ x ph   =>    |- 
 F/ x F/ y ph
 
TheoremnfaldOLD7 29374 If  x is not free in  ph, it is not free in  A. y ph. (Contributed by Mario Carneiro, 24-Sep-2016.)
 |-  F/ y ph   &    |-  ( ph  ->  F/ x ps )   =>    |-  ( ph  ->  F/ x A. y ps )
 
TheoremnfexdOLD7 29375 If  x is not free in  ph, it is not free in  E. y ph. (Contributed by Mario Carneiro, 24-Sep-2016.)
 |-  F/ y ph   &    |-  ( ph  ->  F/ x ps )   =>    |-  ( ph  ->  F/ x E. y ps )
 
Theoremnfa2OLD7 29376 Lemma 24 of [Monk2] p. 114. (Contributed by Mario Carneiro, 24-Sep-2016.)
 |-  F/ x A. y A. x ph
 
TheoremexcomimOLD7 29377 One direction of Theorem 19.11 of [Margaris] p. 89. (Contributed by NM, 5-Aug-1993.) (Revised by Mario Carneiro, 24-Sep-2016.)
 |-  ( E. x E. y ph  ->  E. y E. x ph )
 
TheoremexcomOLD7 29378 Theorem 19.11 of [Margaris] p. 89. (Contributed by NM, 5-Aug-1993.)
 |-  ( E. x E. y ph  <->  E. y E. x ph )
 
Theoremexcom13OLD7 29379 Swap 1st and 3rd existential quantifiers. (Contributed by NM, 9-Mar-1995.)
 |-  ( E. x E. y E. z ph  <->  E. z E. y E. x ph )
 
Theoremexrot3OLD7 29380 Rotate existential quantifiers. (Contributed by NM, 17-Mar-1995.)
 |-  ( E. x E. y E. z ph  <->  E. y E. z E. x ph )
 
Theoremexrot4OLD7 29381 Rotate existential quantifiers twice. (Contributed by NM, 9-Mar-1995.)
 |-  ( E. x E. y E. z E. w ph  <->  E. z E. w E. x E. y ph )
 
TheoremaaanOLD7 29382 Rearrange universal quantifiers. (Contributed by NM, 12-Aug-1993.)
 |-  F/ y ph   &    |-  F/ x ps   =>    |-  ( A. x A. y (
 ph  /\  ps )  <->  (
 A. x ph  /\  A. y ps ) )
 
TheoremeeorOLD7 29383 Rearrange existential quantifiers. (Contributed by NM, 8-Aug-1994.)
 |-  F/ y ph   &    |-  F/ x ps   =>    |-  ( E. x E. y (
 ph  \/  ps )  <->  ( E. x ph  \/  E. y ps ) )
 
Theorempm11.53OLD7 29384* Theorem *11.53 in [WhiteheadRussell] p. 164. (Contributed by Andrew Salmon, 24-May-2011.)
 |-  ( A. x A. y (
 ph  ->  ps )  <->  ( E. x ph 
 ->  A. y ps )
 )
 
Theorem19.12vvOLD7 29385* Special case of 19.12OLD7 29370 where its converse holds. (Contributed by NM, 18-Jul-2001.) (Revised by Andrew Salmon, 11-Jul-2011.)
 |-  ( E. x A. y (
 ph  ->  ps )  <->  A. y E. x ( ph  ->  ps )
 )
 
TheoremeeanOLD7 29386 Rearrange existential quantifiers. (Contributed by NM, 27-Oct-2010.) (Revised by Mario Carneiro, 6-Oct-2016.)
 |-  F/ y ph   &    |-  F/ x ps   =>    |-  ( E. x E. y (
 ph  /\  ps )  <->  ( E. x ph  /\  E. y ps ) )
 
TheoremeeanvOLD7 29387* Rearrange existential quantifiers. (Contributed by NM, 26-Jul-1995.)
 |-  ( E. x E. y (
 ph  /\  ps )  <->  ( E. x ph  /\  E. y ps ) )
 
TheoremeeeanvOLD7 29388* Rearrange existential quantifiers. (Contributed by NM, 26-Jul-1995.) (Proof shortened by Andrew Salmon, 25-May-2011.)
 |-  ( E. x E. y E. z ( ph  /\  ps  /\ 
 ch )  <->  ( E. x ph 
 /\  E. y ps  /\  E. z ch ) )
 
Theoremee4anvOLD7 29389* Rearrange existential quantifiers. (Contributed by NM, 31-Jul-1995.)
 |-  ( E. x E. y E. z E. w ( ph  /\ 
 ps )  <->  ( E. x E. y ph  /\  E. z E. w ps )
 )
 
Theoremax12olem2OLD7 29390* Lemma for ax12oNEW7 29167. Negate the equalities in ax-12 1946, shown as the hypothesis. (Contributed by NM, 24-Dec-2015.)
 |-  ( -.  x  =  y  ->  ( y  =  w 
 ->  A. x  y  =  w ) )   =>    |-  ( -.  x  =  y  ->  ( -.  y  =  z  ->  A. x  -.  y  =  z ) )
 
Theoremax12olem4OLD7 29391* Lemma for ax12oNEW7 29167. Construct an intermediate equivalent to ax-12 1946 from two instances of ax-12 1946. (Contributed by NM, 24-Dec-2015.)
 |-  ( -.  x  =  y  ->  ( y  =  z 
 ->  A. x  y  =  z ) )   &    |-  ( -.  x  =  y  ->  ( y  =  w 
 ->  A. x  y  =  w ) )   =>    |-  ( -.  x  =  y  ->  ( -. 
 A. x  -.  y  =  z  ->  A. x  y  =  z )
 )
 
TheoremhbaeOLD7 29392 All variables are effectively bound in an identical variable specifier. (Contributed by NM, 5-Aug-1993.)
 |-  ( A. x  x  =  y  ->  A. z A. x  x  =  y )
 
TheoremnfaeOLD7 29393 All variables are effectively bound in an identical variable specifier. (Contributed by Mario Carneiro, 11-Aug-2016.)
 |-  F/ z A. x  x  =  y
 
TheoremhbnaeOLD7 29394 All variables are effectively bound in a distinct variable specifier. Lemma L19 in [Megill] p. 446 (p. 14 of the preprint). (Contributed by NM, 5-Aug-1993.)
 |-  ( -.  A. x  x  =  y  ->  A. z  -.  A. x  x  =  y )
 
TheoremnfnaeOLD7 29395 All variables are effectively bound in a distinct variable specifier. (Contributed by Mario Carneiro, 11-Aug-2016.)
 |-  F/ z  -.  A. x  x  =  y
 
TheoremhbnaesOLD7 29396 Rule that applies hbnaeOLD7 29394 to antecedent. (Contributed by NM, 5-Aug-1993.)
 |-  ( A. z  -.  A. x  x  =  y  ->  ph )   =>    |-  ( -.  A. x  x  =  y  ->  ph )
 
TheoremdvelimhOLD7 29397 Version of dvelimOLD7 29423 without any variable restrictions. (Contributed by NM, 1-Oct-2002.)
 |-  ( ph  ->  A. x ph )   &    |-  ( ps  ->  A. z ps )   &    |-  (
 z  =  y  ->  ( ph  <->  ps ) )   =>    |-  ( -.  A. x  x  =  y  ->  ( ps  ->  A. x ps ) )
 
Theoremdral2OLD7 29398 Formula-building lemma for use with the Distinctor Reduction Theorem. Part of Theorem 9.4 of [Megill] p. 448 (p. 16 of preprint). (Contributed by NM, 27-Feb-2005.)
 |-  ( A. x  x  =  y  ->  ( ph  <->  ps ) )   =>    |-  ( A. x  x  =  y  ->  (
 A. z ph  <->  A. z ps )
 )
 
Theoremdrex2OLD7 29399 Formula-building lemma for use with the Distinctor Reduction Theorem. Part of Theorem 9.4 of [Megill] p. 448 (p. 16 of preprint). (Contributed by NM, 27-Feb-2005.)
 |-  ( A. x  x  =  y  ->  ( ph  <->  ps ) )   =>    |-  ( A. x  x  =  y  ->  ( E. z ph  <->  E. z ps )
 )
 
Theoremdrnf2OLD7 29400 Formula-building lemma for use with the Distinctor Reduction Theorem. (Contributed by Mario Carneiro, 4-Oct-2016.)
 |-  ( A. x  x  =  y  ->  ( ph  <->  ps ) )   =>    |-  ( A. x  x  =  y  ->  ( F/ z ph  <->  F/ z ps )
 )
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