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Theorem cbvopabv 4508
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 1712 . 2  |-  F/ z
ph
2 nfv 1712 . 2  |-  F/ w ph
3 nfv 1712 . 2  |-  F/ x ps
4 nfv 1712 . 2  |-  F/ y ps
5 cbvopabv.1 . 2  |-  ( ( x  =  z  /\  y  =  w )  ->  ( ph  <->  ps )
)
61, 2, 3, 4, 5cbvopab 4507 1  |-  { <. x ,  y >.  |  ph }  =  { <. z ,  w >.  |  ps }
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 367    = wceq 1398   {copab 4496
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-rab 2813  df-v 3108  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3784  df-if 3930  df-sn 4017  df-pr 4019  df-op 4023  df-opab 4498
This theorem is referenced by:  cantnf  8103  cantnfOLD  8125  infxpen  8383  axdc2  8820  fpwwe2cbv  8997  fpwwecbv  9011  sylow1  16825  bcth  21937  vitali  22191  lgsquadlem3  23832  lgsquad  23833  ishpg  24332  axcontlem1  24472  eulerpartlemgvv  28582  eulerpart  28588  cvmlift2lem13  29027  pellex  31013  aomclem8  31249
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