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Theorem cbvopabv 3960
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 1491 . 2  |-  F/ z
ph
2 nfv 1491 . 2  |-  F/ w ph
3 nfv 1491 . 2  |-  F/ x ps
4 nfv 1491 . 2  |-  F/ y ps
5 cbvopabv.1 . 2  |-  ( ( x  =  z  /\  y  =  w )  ->  ( ph  <->  ps )
)
61, 2, 3, 4, 5cbvopab 3959 1  |-  { <. x ,  y >.  |  ph }  =  { <. z ,  w >.  |  ps }
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    <-> wb 104    = wceq 1314   {copab 3948
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 681  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-10 1466  ax-11 1467  ax-i12 1468  ax-bndl 1469  ax-4 1470  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097
This theorem depends on definitions:  df-bi 116  df-3an 947  df-tru 1317  df-nf 1420  df-sb 1719  df-clab 2102  df-cleq 2108  df-clel 2111  df-nfc 2244  df-v 2659  df-un 3041  df-sn 3499  df-pr 3500  df-op 3502  df-opab 3950
This theorem is referenced by: (None)
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