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Theorem cbvopabv 4269
Description: Rule used to change bound variables in an ordered-pair class abstraction, using implicit substitution. (Contributed by NM, 15-Oct-1996.)
Hypothesis
Ref Expression
cbvopabv.1  |-  ( ( x  =  z  /\  y  =  w )  ->  ( ph  <->  ps )
)
Assertion
Ref Expression
cbvopabv  |-  { <. x ,  y >.  |  ph }  =  { <. z ,  w >.  |  ps }
Distinct variable groups:    x, y,
z, w    ph, z, w    ps, x, y
Allowed substitution hints:    ph( x, y)    ps( z, w)

Proof of Theorem cbvopabv
StepHypRef Expression
1 nfv 1629 . 2  |-  F/ z
ph
2 nfv 1629 . 2  |-  F/ w ph
3 nfv 1629 . 2  |-  F/ x ps
4 nfv 1629 . 2  |-  F/ y ps
5 cbvopabv.1 . 2  |-  ( ( x  =  z  /\  y  =  w )  ->  ( ph  <->  ps )
)
61, 2, 3, 4, 5cbvopab 4268 1  |-  { <. x ,  y >.  |  ph }  =  { <. z ,  w >.  |  ps }
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359    = wceq 1652   {copab 4257
This theorem is referenced by:  cantnf  7641  infxpen  7888  axdc2  8321  fpwwe2cbv  8497  fpwwecbv  8511  sylow1  15229  bcth  19274  vitali  19497  lgsquadlem3  21132  lgsquad  21133  cvmlift2lem13  24994  axcontlem1  25895  pellex  26889  aomclem8  27127
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-rab 2706  df-v 2950  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-sn 3812  df-pr 3813  df-op 3815  df-opab 4259
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