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Theorem List for Metamath Proof Explorer - 29601-29700   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
TheoremdvelimfwAUX7 29601* Version of dvelimvNEW7 29399. (Contributed by Mario Carneiro, 6-Oct-2016.) (Revised by NM, 22-May-2018.)
 |-  F/ x ph   &    |-  F/ z ps   &    |-  ( z  =  y  ->  ( ph  <->  ps ) )   =>    |-  ( -.  A. x  x  =  y  ->  F/ x ps )
 
TheoremnfsbwAUX7 29602* If  z is not free in  ph, it is not free in  [ y  /  x ] ph. (Contributed by Mario Carneiro, 11-Aug-2016.) (Revised by NM, 22-May-2018.)
 |-  F/ z ph   =>    |- 
 F/ z [ y  /  x ] ph
 
Theoremax7w15lemxAUX7 29603* Lemma for ax7w14AUX7 29607. (Contributed by NM, 23-May-2018.)
 |-  F/ x ph   &    |-  F/ y ph   =>    |-  ( -.  A. x  x  =  y  ->  ( A. x A. y [ x  /  u ] [ y  /  v ] ph  ->  A. s A. t [
 s  /  u ] [ t  /  v ] ph ) )
 
Theoremax7w15lemAUX7 29604* Lemma for ax7w15AUX7 29605. (Contributed by NM, 22-May-2018.)
 |-  F/ x ph   &    |-  F/ y ph   &    |-  ( ph  ->  F/ x ps )   &    |-  ( ph  ->  F/ y ps )   =>    |-  ( ( ph  /\  -.  A. x  x  =  y )  ->  ( A. x A. y [ x  /  u ] [ y  /  v ] ps  <->  A. s A. t [ s  /  u ] [ t  /  v ] ps ) )
 
Theoremax7w15AUX7 29605* Special case of ax-7 1749 derived from ax-7v 29379. (Contributed by NM, 22-May-2018.)
 |-  F/ x ph   &    |-  F/ y ph   &    |-  ( ph  ->  F/ x ps )   &    |-  ( ph  ->  F/ y ps )   =>    |-  ( ph  ->  ( A. x A. y [ x  /  u ] [
 y  /  v ] ps  ->  A. y A. x [ x  /  u ] [ y  /  v ] ps ) )
 
Theoremax7w14lemAUX7 29606* Lemma for ax7w14AUX7 29607. (Contributed by NM, 22-May-2018.)
 |-  F/ x ph   &    |-  F/ y ph   =>    |-  ( -.  A. x  x  =  y  ->  ( A. x A. y [ x  /  u ] [ y  /  v ] ph  <->  A. s A. t [ s  /  u ] [ t  /  v ] ph ) )
 
Theoremax7w14AUX7 29607* Special case of ax-7 1749 derived from ax-7v 29379. (Contributed by NM, 22-May-2018.)
 |-  F/ x ph   &    |-  F/ y ph   =>    |-  ( A. x A. y [ x  /  u ] [
 y  /  v ] ph  ->  A. y A. x [ x  /  u ] [ y  /  v ] ph )
 
Theoremax7w12AUX7 29608* Special case of ax-7 1749 derived from ax-7v 29379. (Contributed by NM, 21-May-2018.)
 |-  ( A. x A. y [ x  /  u ] [
 y  /  v ] ph  ->  A. y A. x [ x  /  u ] [ y  /  v ] ph )
 
Theoremelsb34AUX7 29609* Substitution applied to an atomic membership wff. (Contributed by NM, 21-May-2018.)
 |-  ( [ x  /  u ] [ y  /  v ] u  e.  v  <->  x  e.  y )
 
Theoremax7w13AUX7 29610 Special case of ax-7 1749 derived from ax-7v 29379. Example illustrating use of ax7w12AUX7 29608. (Contributed by NM, 21-May-2018.)
 |-  ( A. x A. y  x  e.  y  ->  A. y A. x  x  e.  y )
 
Theoremax7w16AUX7 29611 Special case of ax-7 1749 derived from ax-7v 29379. This is the "easy" direction. The other direction is still conjectured. (Contributed by NM, 24-May-2018.)
 |-  ( A. z A. x  x  =  y  ->  A. x A. z  x  =  y )
 
Theoremax7w17lem1AUX7 29612* Lemma for ax7w17lem2AUX7 29613. (Contributed by NM, 24-May-2018.)
 |-  ( [ x  /  u ] [ z  /  v ] u  =  y  <->  x  =  y )
 
Theoremax7w17lem2AUX7 29613* Experimental lemma with the goal of deriving hbae 2040 from ax-7v 29379. Looks like a dead end because we apparently need hbae 2040 to eliminate the first 2 hypotheses for those  ph that satisfy the 3rd and 4th hypotheses. (Contributed by NM, 24-May-2018.)
 |-  F/ x ph   &    |-  F/ z ph   &    |-  ( ph  ->  F/ x  u  =  y )   &    |-  ( ph  ->  F/ z  u  =  y )   =>    |-  ( ph  ->  (
 A. x A. z  x  =  y  ->  A. z A. x  x  =  y ) )
 
Theoremax7w18AUX7 29614* Weak version of ax-7 1749 not requiring ax-7 1749. (Contributed by NM, 24-May-2018.)
 |-  ( A. u  u  =  v  ->  ( A. x A. y ph  ->  A. y A. x ph ) )
 
19.26.1.2  Theorems derived from ax-7 (suffix OLD7)
 
Axiomax-7OLD7 29615 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 1733) 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 29616 Swap quantifiers in an antecedent. (Contributed by NM, 5-Aug-1993.)
 |-  ( A. x A. y ph  ->  ps )   =>    |-  ( A. y A. x ph  ->  ps )
 
TheoremhbalOLD7 29617 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 29618 Theorem 19.5 of [Margaris] p. 89. (Contributed by NM, 5-Aug-1993.)
 |-  ( A. x A. y ph  <->  A. y A. x ph )
 
Theoremalrot3OLD7 29619 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 29620 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 29621 Deduction form of bound-variable hypothesis builder hbalOLD7 29617. (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 29622 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 29623 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 29638 and r19.12sn 3864. (Contributed by NM, 5-Aug-1993.)
 |-  ( E. x A. y ph  ->  A. y E. x ph )
 
TheoremnfalOLD7 29624 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 29625 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 29626 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 29627 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 29628 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 29629 Lemma 24 of [Monk2] p. 114. (Contributed by Mario Carneiro, 24-Sep-2016.)
 |-  F/ x A. y A. x ph
 
TheoremexcomimOLD7 29630 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 29631 Theorem 19.11 of [Margaris] p. 89. (Contributed by NM, 5-Aug-1993.)
 |-  ( E. x E. y ph  <->  E. y E. x ph )
 
Theoremexcom13OLD7 29632 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 29633 Rotate existential quantifiers. (Contributed by NM, 17-Mar-1995.)
 |-  ( E. x E. y E. z ph  <->  E. y E. z E. x ph )
 
Theoremexrot4OLD7 29634 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 29635 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 29636 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 29637* 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 29638* Special case of 19.12OLD7 29623 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 29639 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 29640* Rearrange existential quantifiers. (Contributed by NM, 26-Jul-1995.)
 |-  ( E. x E. y (
 ph  /\  ps )  <->  ( E. x ph  /\  E. y ps ) )
 
TheoremeeeanvOLD7 29641* 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 29642* 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 29643* Lemma for ax12oNEW7 29398. Negate the equalities in ax-12 1950, 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 29644* Lemma for ax12oNEW7 29398. Construct an intermediate equivalent to ax-12 1950 from two instances of ax-12 1950. (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 29645 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 29646 All variables are effectively bound in an identical variable specifier. (Contributed by Mario Carneiro, 11-Aug-2016.)
 |-  F/ z A. x  x  =  y
 
TheoremhbnaeOLD7 29647 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 29648 All variables are effectively bound in a distinct variable specifier. (Contributed by Mario Carneiro, 11-Aug-2016.)
 |-  F/ z  -.  A. x  x  =  y
 
TheoremhbnaesOLD7 29649 Rule that applies hbnaeOLD7 29647 to antecedent. (Contributed by NM, 5-Aug-1993.)
 |-  ( A. z  -.  A. x  x  =  y  ->  ph )   =>    |-  ( -.  A. x  x  =  y  ->  ph )
 
TheoremdvelimhOLD7 29650 Version of dvelimOLD7 29676 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 29651 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 29652 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 29653 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 29654 Variation on nfaldOLD7 29627 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 29655 Variation on nfexdOLD7 29628 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 29656 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 29657 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 29658 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 29659 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 29660 Rule used to change bound variables, using implicit substitution, that does not use ax-12o 2218. (Contributed by NM, 5-Aug-1993.)
 |-  F/ y ph   &    |-  F/ x ps   &    |-  ( x  =  y  ->  (
 ph  ->  ps ) )   =>    |-  ( A. x ph 
 ->  A. y ps )
 
Theoremcbv3hOLD7 29661 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 29662 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 29663 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 29664 Version of dvelimvNEW7 29399 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 29665* 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 29666* 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 29667* 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 29668* 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 29669* 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 29670* 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 29671* Deduction used to change bound variables, using implicit substitution, particularly useful in conjunction with dvelimOLD7 29676. (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 29672* Deduction used to change bound variables, using implicit substitution, particularly useful in conjunction with dvelimOLD7 29676. (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 29673* 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 29674* 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 29675* 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 29676* 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 29688.

Other variants of this theorem are dvelimhOLD7 29650 (with no distinct variable restrictions), dvelimhwNEW7 29392 (that avoids ax-12 1950), and dvelimALT 2209 (that avoids ax-10 2216). (Contributed by NM, 23-Nov-1994.)

 |-  ( ph  ->  A. x ph )   &    |-  (
 z  =  y  ->  ( ph  <->  ps ) )   =>    |-  ( -.  A. x  x  =  y  ->  ( ps  ->  A. x ps ) )
 
TheoremdvelimnfOLD7 29677* Version of dvelimOLD7 29676 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 29678 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 29679 Substitution does not change a distinctor. (Contributed by NM, 5-Aug-1993.)
 |-  ( [ w  /  z ]  -.  A. x  x  =  y  <->  -.  A. x  x  =  y )
 
Theoremax16ALT2OLD7 29680* Alternate proof of ax16NEW7 29486. (Contributed by NM, 8-Nov-2006.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  ( A. x  x  =  y  ->  ( ph  ->  A. x ph ) )
 
Theorema16gALTOLD7 29681* A generalization of axiom ax-16 2220. Alternate proof of a16gNEW7 29483 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 ) )
 
Theoremnfsb4tOLD7 29682 A variable not free remains so after substitution with a distinct variable (closed form of nfsb4OLD7 29684). (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 29683* Weak version of nfsb4t 2154. Still uses ax-7OLD7 29615 via nfaldOLD7 29627. (Contributed by NM, 25-Nov-2017.)
 |-  ( A. x F/ z ph  ->  ( -.  A. z  z  =  y  ->  F/ z [ y  /  x ] ph ) )
 
Theoremnfsb4OLD7 29684 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 )
 
TheoremnfsbOLD7 29685* 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
 
TheoremhbsbOLD7 29686* 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 )
 
TheoremnfsbdOLD7 29687* Deduction version of nfsbOLD7 29685. (Contributed by NM, 15-Feb-2013.)
 |-  F/ x ph   &    |-  ( ph  ->  F/ z ps )   =>    |-  ( ph  ->  F/ z [ y  /  x ] ps )
 
TheoremdvelimdfOLD7 29688 Deduction form of dvelimfOLD7 29664. This version may be useful if we want to avoid ax-17 1626 and use ax-16 2220 instead. (Contributed by NM, 7-Apr-2004.) (Revised by Mario Carneiro, 6-Oct-2016.)
 |-  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 )
 )
 
Theoremsbco2OLD7 29689 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 )
 
Theoremsbco2dOLD7 29690 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 ) )
 
Theoremsbco3OLD7 29691 A composition law for substitution. (Contributed by NM, 5-Aug-1993.)
 |-  ( [ z  /  y ] [ y  /  x ] ph  <->  [ z  /  x ] [ x  /  y ] ph )
 
TheoremsbcomOLD7 29692 A commutativity law for substitution. (Contributed by NM, 27-May-1997.)
 |-  ( [ y  /  z ] [ y  /  x ] ph  <->  [ y  /  x ] [ y  /  z ] ph )
 
Theoremsb8OLD7 29693 Substitution of variable in universal quantifier. (Contributed by NM, 5-Aug-1993.) (Revised by Mario Carneiro, 6-Oct-2016.)
 |-  F/ y ph   =>    |-  ( A. x ph  <->  A. y [ y  /  x ] ph )
 
Theoremsb8eOLD7 29694 Substitution of variable in existential quantifier. (Contributed by NM, 12-Aug-1993.) (Revised by Mario Carneiro, 6-Oct-2016.)
 |-  F/ y ph   =>    |-  ( E. x ph  <->  E. y [ y  /  x ] ph )
 
Theoremsb9iOLD7 29695 Commutation of quantification and substitution variables. (Contributed by NM, 5-Aug-1993.)
 |-  ( A. x [ x  /  y ] ph  ->  A. y [ y  /  x ] ph )
 
Theoremsb9OLD7 29696 Commutation of quantification and substitution variables. (Contributed by NM, 5-Aug-1993.)
 |-  ( A. x [ x  /  y ] ph  <->  A. y [ y  /  x ] ph )
 
TheoremsbhbOLD7 29697* 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 ) )
 
Theorem2sb5rfOLD7 29698* 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 ) )
 
Theorem2sb6rfOLD7 29699* 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 )
 )
 
Theoremdfsb7OLD7 29700* 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 sb5NEW7 29535, 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 sb7hOLD7 29702 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 )
 ) )
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