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Theorem List for Metamath Proof Explorer - 29601-29700   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremsb6xNEW7 29601 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 29602 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 29603 A substitution into a theorem remains true. (See chvarNEW7 29592 and chvarvNEW7 29598 for versions using implicit substitution.) (Contributed by NM, 21-Jan-2004.) (Proof shortened by Andrew Salmon, 25-May-2011.)
 |-  ph   =>    |- 
 [ y  /  x ] ph
 
TheoremsbiedvNEW7 29604* Conversion of implicit substitution to explicit substitution (deduction version of sbieNEW7 29516). (Contributed by NM, 7-Jan-2017.)
 |-  (
 ( ph  /\  x  =  y )  ->  ( ps 
 <->  ch ) )   =>    |-  ( ph  ->  ( [ y  /  x ] ps  <->  ch ) )
 
Theoremsb6fNEW7 29605 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 29606 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 29607* Alternate proof of ax16NEW7 29524. (Contributed by NM, 17-May-2008.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  ( A. x  x  =  y  ->  ( ph  ->  A. x ph ) )
 
Theoremsb3NEW7 29608 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 29609 An alternate definition of proper substitution that, like df-sb 1639, 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 29610 An alternate definition of proper substitution df-sb 1639 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 29611 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 29612 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 29613 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 29614 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 29615 Weak version of ax-7 1720 not requiring ax-7 1720. (Contributed by NM, 9-Oct-2017.)
 |-  ( A. x  x  =  y  ->  ( A. x A. y ph  ->  A. y A. x ph ) )
 
Theoremax7w1hAUX7 29616 Weak version of hbal 1722 not requiring ax-7 1720. (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 29617 Weak version of hbae 1906 not requiring ax-7 1720. (Contributed by NM, 29-Oct-2017.)
 |-  ( A. x  x  =  y  ->  A. z  x  =  y )
 
Theoremhbaew4AUX7 29618 Weak version of hbae 1906 not requiring ax-7 1720. (Contributed by NM, 30-Oct-2017.)
 |-  ( A. x  x  =  z  ->  ( A. x  x  =  y  ->  A. z A. x  x  =  y ) )
 
Theoremhbaew5AUX7 29619* Weak version of hbae 1906 not requiring ax-7 1720. (Contributed by NM, 30-Oct-2017.)
 |-  ( A. u  u  =  v  ->  ( A. x  x  =  y  ->  A. z A. x  x  =  y ) )
 
Theoremax7w2AUX7 29620 Special case of ax-7 1720. (Contributed by NM, 9-Oct-2017.)
 |-  ( A. x A. y [
 y  /  x ] ph  ->  A. y A. x [ y  /  x ] ph )
 
Theoremax7w3AUX7 29621 Special case of ax-7 1720. (Contributed by NM, 12-Oct-2017.)
 |-  ( A. x A. y  x  =  y  ->  A. y A. x  x  =  y )
 
Theoremax7w6AUX7 29622 Version of sb9i 2047 with the usage of ax-7 1720 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 29623 Special case of ax-7 1720. (Contributed by NM, 12-Oct-2017.)
 |-  ( A. x A. y  -.  x  =  y  ->  A. y A. x  -.  x  =  y )
 
Theoremax7w8AUX7 29624 Special case of ax-7 1720. (Contributed by NM, 13-Oct-2017.)
 |-  ( ph  ->  A. x ph )   =>    |-  ( A. x A. y ph  ->  A. y A. x ph )
 
Theoremax7nfAUX7 29625 Special case of ax-7 1720. (Contributed by NM, 23-Nov-2017.)
 |-  F/ x ph   =>    |-  ( A. x A. y ph  ->  A. y A. x ph )
 
Theoremax7w7tAUX7 29626 Special case of ax-7 1720. (Contributed by NM, 23-Nov-2017.)
 |-  ( A. y ( ph  ->  A. x ph )  ->  ( A. x A. y ph  ->  A. y A. x ph ) )
 
Theoremax7wnftAUX7 29627 Special case of ax-7 1720. (Contributed by NM, 23-Nov-2017.)
 |-  ( A. y F/ x ph  ->  ( A. x A. y ph  ->  A. y A. x ph ) )
 
Theoremax7w4AUX7 29628 Remove quantifier from ax-7 1720. (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 29629 Same as hbal 1722 with explicit ax-7 1720 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 29630 Special case of ax-7 1720 proved from ax-7v 29419. (Contributed by NM, 28-Nov-2017.)
 |-  ( A. x A. y ( x  =  y  /\  x  =  z )  ->  A. y A. x ( x  =  y  /\  x  =  z
 ) )
 
Theoremalcomw9bAUX7 29631 Special case of alcom 1723 proved from ax-7v 29419. (Contributed by NM, 28-Nov-2017.)
 |-  ( A. x A. y ( x  =  y  /\  x  =  z )  <->  A. y A. x ( x  =  y  /\  x  =  z )
 )
 
18.27.1.2  Theorems derived from ax-7 (suffix OLD7)
 
Axiomax-7OLD7 29632 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 1704) 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 29633 Swap quantifiers in an antecedent. (Contributed by NM, 5-Aug-1993.)
 |-  ( A. x A. y ph  ->  ps )   =>    |-  ( A. y A. x ph  ->  ps )
 
TheoremhbalOLD7 29634 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 29635 Theorem 19.5 of [Margaris] p. 89. (Contributed by NM, 5-Aug-1993.)
 |-  ( A. x A. y ph  <->  A. y A. x ph )
 
Theoremalrot3OLD7 29636 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 29637 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 29638 Deduction form of bound-variable hypothesis builder hbalOLD7 29634. (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 29639 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 29640 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 29655 and r19.12sn 3709. (Contributed by NM, 5-Aug-1993.)
 |-  ( E. x A. y ph  ->  A. y E. x ph )
 
TheoremnfalOLD7 29641 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 29642 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 29643 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 29644 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 29645 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 29646 Lemma 24 of [Monk2] p. 114. (Contributed by Mario Carneiro, 24-Sep-2016.)
 |-  F/ x A. y A. x ph
 
TheoremexcomimOLD7 29647 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 29648 Theorem 19.11 of [Margaris] p. 89. (Contributed by NM, 5-Aug-1993.)
 |-  ( E. x E. y ph  <->  E. y E. x ph )
 
Theoremexcom13OLD7 29649 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 29650 Rotate existential quantifiers. (Contributed by NM, 17-Mar-1995.)
 |-  ( E. x E. y E. z ph  <->  E. y E. z E. x ph )
 
Theoremexrot4OLD7 29651 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 29652 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 29653 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 29654* 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 29655* Special case of 19.12OLD7 29640 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 29656 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 29657* Rearrange existential quantifiers. (Contributed by NM, 26-Jul-1995.)
 |-  ( E. x E. y (
 ph  /\  ps )  <->  ( E. x ph  /\  E. y ps ) )
 
TheoremeeeanvOLD7 29658* 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 29659* 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 29660* Lemma for ax12oNEW7 29438. Negate the equalities in ax-12 1878, 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 29661* Lemma for ax12oNEW7 29438. Construct an intermediate equivalent to ax-12 1878 from two instances of ax-12 1878. (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 29662 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 29663 All variables are effectively bound in an identical variable specifier. (Contributed by Mario Carneiro, 11-Aug-2016.)
 |-  F/ z A. x  x  =  y
 
TheoremhbnaeOLD7 29664 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 29665 All variables are effectively bound in a distinct variable specifier. (Contributed by Mario Carneiro, 11-Aug-2016.)
 |-  F/ z  -.  A. x  x  =  y
 
TheoremhbnaesOLD7 29666 Rule that applies hbnaeOLD7 29664 to antecedent. (Contributed by NM, 5-Aug-1993.)
 |-  ( A. z  -.  A. x  x  =  y  ->  ph )   =>    |-  ( -.  A. x  x  =  y  ->  ph )
 
TheoremdvelimhOLD7 29667 Version of dvelimOLD7 29693 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 29668 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 29669 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 29670 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 )
 )
 
Theoremnfald2OLD7 29671 Variation on nfaldOLD7 29644 which adds the hypothesis that  x and  y are distinct in the inner subproof. (Contributed by Mario Carneiro, 8-Oct-2016.)
 |-  F/ y ph   &    |-  ( ( ph  /\ 
 -.  A. x  x  =  y )  ->  F/ x ps )   =>    |-  ( ph  ->  F/ x A. y ps )
 
Theoremnfexd2OLD7 29672 Variation on nfexdOLD7 29645 which adds the hypothesis that  x and  y are distinct in the inner subproof. (Contributed by Mario Carneiro, 8-Oct-2016.)
 |-  F/ y ph   &    |-  ( ( ph  /\ 
 -.  A. x  x  =  y )  ->  F/ x ps )   =>    |-  ( ph  ->  F/ x E. y ps )
 
Theoremcbv1hOLD7 29673 Rule used to change bound variables, using implicit substitution. (Contributed by NM, 5-Aug-1993.)
 |-  ( 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 ) )
 
Theoremcbv1OLD7 29674 Rule used to change bound variables, using implicit substitution. (Contributed by NM, 5-Aug-1993.) (Revised by Mario Carneiro, 3-Oct-2016.)
 |-  ( ph  ->  F/ y ps )   &    |-  ( ph  ->  F/ x ch )   &    |-  ( ph  ->  ( x  =  y  ->  ( ps  ->  ch ) ) )   =>    |-  ( A. x A. y ph  ->  ( A. x ps  ->  A. y ch )
 )
 
Theoremcbv2hOLD7 29675 Rule used to change bound variables, using implicit substitution. (Contributed by NM, 5-Aug-1993.)
 |-  ( 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 )
 )
 
Theoremcbv2OLD7 29676 Rule used to change bound variables, using implicit substitution. (Contributed by NM, 5-Aug-1993.) (Revised by Mario Carneiro, 3-Oct-2016.)
 |-  ( ph  ->  F/ y ps )   &    |-  ( ph  ->  F/ x ch )   &    |-  ( ph  ->  ( x  =  y  ->  ( ps  <->  ch ) ) )   =>    |-  ( A. x A. y ph  ->  ( A. x ps  <->  A. y ch )
 )
 
Theoremcbv3OLD7 29677 Rule used to change bound variables, using implicit substitution, that does not use ax-12o 2094. (Contributed by NM, 5-Aug-1993.)
 |-  F/ y ph   &    |-  F/ x ps   &    |-  ( x  =  y  ->  (
 ph  ->  ps ) )   =>    |-  ( A. x ph 
 ->  A. y ps )
 
Theoremcbv3hOLD7 29678 Rule used to change bound variables, using implicit substitution. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Andrew Salmon, 25-May-2011.) (Proof modification is discouraged.)
 |-  ( ph  ->  A. y ph )   &    |-  ( ps  ->  A. x ps )   &    |-  ( x  =  y  ->  (
 ph  ->  ps ) )   =>    |-  ( A. x ph 
 ->  A. y ps )
 
TheoremcbvalOLD7 29679 Rule used to change bound variables, using implicit substitution. (Contributed by NM, 5-Aug-1993.) (Revised by Mario Carneiro, 3-Oct-2016.)
 |-  F/ y ph   &    |-  F/ x ps   &    |-  ( x  =  y  ->  (
 ph 
 <->  ps ) )   =>    |-  ( A. x ph  <->  A. y ps )
 
TheoremcbvexOLD7 29680 Rule used to change bound variables, using implicit substitution. (Contributed by NM, 5-Aug-1993.)
 |-  F/ y ph   &    |-  F/ x ps   &    |-  ( x  =  y  ->  (
 ph 
 <->  ps ) )   =>    |-  ( E. x ph  <->  E. y ps )
 
TheoremdvelimfOLD7 29681 Version of dvelimvNEW7 29439 without any variable restrictions. (Contributed by NM, 1-Oct-2002.) (Revised by Mario Carneiro, 6-Oct-2016.)
 |-  F/ x ph   &    |-  F/ z ps   &    |-  ( z  =  y  ->  ( ph  <->  ps ) )   =>    |-  ( -.  A. x  x  =  y  ->  F/ x ps )
 
TheoremcbvalvOLD7 29682* Rule used to change bound variables, using implicit substitution. (Contributed by NM, 5-Aug-1993.)
 |-  ( x  =  y  ->  (
 ph 
 <->  ps ) )   =>    |-  ( A. x ph  <->  A. y ps )
 
TheoremcbvexvOLD7 29683* Rule used to change bound variables, using implicit substitution. (Contributed by NM, 5-Aug-1993.)
 |-  ( x  =  y  ->  (
 ph 
 <->  ps ) )   =>    |-  ( E. x ph  <->  E. y ps )
 
Theoremcbval2OLD7 29684* Rule used to change bound variables, using implicit substitution. (Contributed by NM, 22-Dec-2003.) (Revised by Mario Carneiro, 6-Oct-2016.)
 |-  F/ z ph   &    |-  F/ w ph   &    |-  F/ x ps   &    |-  F/ y ps   &    |-  ( ( x  =  z  /\  y  =  w )  ->  ( ph 
 <->  ps ) )   =>    |-  ( A. x A. y ph  <->  A. z A. w ps )
 
Theoremcbvex2OLD7 29685* Rule used to change bound variables, using implicit substitution. (Contributed by NM, 14-Sep-2003.) (Revised by Mario Carneiro, 6-Oct-2016.)
 |-  F/ z ph   &    |-  F/ w ph   &    |-  F/ x ps   &    |-  F/ y ps   &    |-  ( ( x  =  z  /\  y  =  w )  ->  ( ph 
 <->  ps ) )   =>    |-  ( E. x E. y ph  <->  E. z E. w ps )
 
Theoremcbval2vOLD7 29686* Rule used to change bound variables, using implicit substitution. (Contributed by NM, 4-Feb-2005.)
 |-  (
 ( x  =  z 
 /\  y  =  w )  ->  ( ph  <->  ps ) )   =>    |-  ( A. x A. y ph  <->  A. z A. w ps )
 
Theoremcbvex2vOLD7 29687* Rule used to change bound variables, using implicit substitution. (Contributed by NM, 26-Jul-1995.)
 |-  (
 ( x  =  z 
 /\  y  =  w )  ->  ( ph  <->  ps ) )   =>    |-  ( E. x E. y ph  <->  E. z E. w ps )
 
TheoremcbvaldOLD7 29688* Deduction used to change bound variables, using implicit substitution, particularly useful in conjunction with dvelimOLD7 29693. (Contributed by NM, 2-Jan-2002.) (Revised by Mario Carneiro, 6-Oct-2016.)
 |-  F/ y ph   &    |-  ( ph  ->  F/ y ps )   &    |-  ( ph  ->  ( x  =  y  ->  ( ps  <->  ch ) ) )   =>    |-  ( ph  ->  (
 A. x ps  <->  A. y ch )
 )
 
TheoremcbvexdOLD7 29689* Deduction used to change bound variables, using implicit substitution, particularly useful in conjunction with dvelimOLD7 29693. (Contributed by NM, 2-Jan-2002.) (Revised by Mario Carneiro, 6-Oct-2016.)
 |-  F/ y ph   &    |-  ( ph  ->  F/ y ps )   &    |-  ( ph  ->  ( x  =  y  ->  ( ps  <->  ch ) ) )   =>    |-  ( ph  ->  ( E. x ps  <->  E. y ch )
 )
 
TheoremcbvaldvaOLD7 29690* Rule used to change the bound variable in a universal quantifier with implicit substitution. Deduction form. (Contributed by David Moews, 1-May-2017.)
 |-  (
 ( ph  /\  x  =  y )  ->  ( ps 
 <->  ch ) )   =>    |-  ( ph  ->  (
 A. x ps  <->  A. y ch )
 )
 
TheoremcbvexdvaOLD7 29691* Rule used to change the bound variable in an existential quantifier with implicit substitution. Deduction form. (Contributed by David Moews, 1-May-2017.)
 |-  (
 ( ph  /\  x  =  y )  ->  ( ps 
 <->  ch ) )   =>    |-  ( ph  ->  ( E. x ps  <->  E. y ch )
 )
 
Theoremcbvex4vOLD7 29692* Rule used to change bound variables, using implicit substitution. (Contributed by NM, 26-Jul-1995.)
 |-  (
 ( x  =  v 
 /\  y  =  u )  ->  ( ph  <->  ps ) )   &    |-  ( ( z  =  f  /\  w  =  g )  ->  ( ps 
 <->  ch ) )   =>    |-  ( E. x E. y E. z E. w ph  <->  E. v E. u E. f E. g ch )
 
TheoremdvelimOLD7 29693* This theorem can be used to eliminate a distinct variable restriction on  x and  z and replace it with the "distinctor"  -.  A. x x  =  y as an antecedent.  ph normally has  z free and can be read  ph ( z ), and  ps substitutes  y for  z and can be read  ph ( y ). We don't require that 
x and  y be distinct: if they aren't, the distinctor will become false (in multiple-element domains of discourse) and "protect" the consequent.

To obtain a closed-theorem form of this inference, prefix the hypotheses with  A. x A. z, conjoin them, and apply dvelimdfOLD7 29705.

Other variants of this theorem are dvelimhOLD7 29667 (with no distinct variable restrictions), dvelimhwNEW7 29432 (that avoids ax-12 1878), and dvelimALTOLD7 (future) (that avoids ax-10 2092). (Contributed by NM, 23-Nov-1994.)

 |-  ( ph  ->  A. x ph )   &    |-  (
 z  =  y  ->  ( ph  <->  ps ) )   =>    |-  ( -.  A. x  x  =  y  ->  ( ps  ->  A. x ps ) )
 
TheoremdvelimnfOLD7 29694* Version of dvelimOLD7 29693 using "not free" notation. (Contributed by Mario Carneiro, 9-Oct-2016.)
 |-  F/ x ph   &    |-  ( z  =  y  ->  ( ph  <->  ps ) )   =>    |-  ( -.  A. x  x  =  y  ->  F/ x ps )
 
Theoremsbequ5OLD7 29695 Substitution does not change an identical variable specifier. (Contributed by NM, 5-Aug-1993.)
 |-  ( [ w  /  z ] A. x  x  =  y  <->  A. x  x  =  y )
 
Theoremsbequ6OLD7 29696 Substitution does not change a distinctor. (Contributed by NM, 5-Aug-1993.)
 |-  ( [ w  /  z ]  -.  A. x  x  =  y  <->  -.  A. x  x  =  y )
 
Theoremax16ALT2OLD7 29697* Alternate proof of ax16NEW7 29524. (Contributed by NM, 8-Nov-2006.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  ( A. x  x  =  y  ->  ( ph  ->  A. x ph ) )
 
Theorema16gALTOLD7 29698* A generalization of axiom ax-16 2096. Alternate proof of a16gNEW7 29521 that uses df-sb 1639. (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 ) )
 
Theoremnfsb4tOLD7 29699 A variable not free remains so after substitution with a distinct variable (closed form of nfsb4OLD7 29701). (Contributed by NM, 7-Apr-2004.) (Revised by Mario Carneiro, 4-Oct-2016.)
 |-  ( A. x F/ z ph  ->  ( -.  A. z  z  =  y  ->  F/ z [ y  /  x ] ph ) )
 
Theoremnfsb4tw2AUXOLD7 29700* Weak version of nfsb4t 2033. Still uses ax-7OLD7 29632 via nfaldOLD7 29644. (Contributed by NM, 25-Nov-2017.)
 |-  ( A. x F/ z ph  ->  ( -.  A. z  z  =  y  ->  F/ z [ y  /  x ] ph ) )
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