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Theorem cbvoprab3v 5606
Description: Rule used to change the third bound variable in an operation abstraction, using implicit substitution. (Contributed by NM, 8-Oct-2004.) (Revised by David Abernethy, 19-Jun-2012.)
Hypothesis
Ref Expression
cbvoprab3v.1  |-  ( z  =  w  ->  ( ph 
<->  ps ) )
Assertion
Ref Expression
cbvoprab3v  |-  { <. <.
x ,  y >. ,  z >.  |  ph }  =  { <. <. x ,  y >. ,  w >.  |  ps }
Distinct variable groups:    x, z, w   
y, z, w    ph, w    ps, z
Allowed substitution hints:    ph( x, y, z)    ps( x, y, w)

Proof of Theorem cbvoprab3v
StepHypRef Expression
1 nfv 1462 . 2  |-  F/ w ph
2 nfv 1462 . 2  |-  F/ z ps
3 cbvoprab3v.1 . 2  |-  ( z  =  w  ->  ( ph 
<->  ps ) )
41, 2, 3cbvoprab3 5605 1  |-  { <. <.
x ,  y >. ,  z >.  |  ph }  =  { <. <. x ,  y >. ,  w >.  |  ps }
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 103    = wceq 1285   {coprab 5538
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-14 1446  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2064  ax-sep 3898  ax-pow 3950  ax-pr 3966
This theorem depends on definitions:  df-bi 115  df-3an 922  df-tru 1288  df-nf 1391  df-sb 1687  df-clab 2069  df-cleq 2075  df-clel 2078  df-nfc 2209  df-v 2604  df-un 2978  df-in 2980  df-ss 2987  df-pw 3386  df-sn 3406  df-pr 3407  df-op 3409  df-opab 3842  df-oprab 5541
This theorem is referenced by: (None)
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