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Theorem cbvoprab3v 6116
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 1626 . 2  |-  F/ w ph
2 nfv 1626 . 2  |-  F/ z ps
3 cbvoprab3v.1 . 2  |-  ( z  =  w  ->  ( ph 
<->  ps ) )
41, 2, 3cbvoprab3 6115 1  |-  { <. <.
x ,  y >. ,  z >.  |  ph }  =  { <. <. x ,  y >. ,  w >.  |  ps }
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    = wceq 1649   {coprab 6049
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2393  ax-sep 4298  ax-nul 4306  ax-pr 4371
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-clab 2399  df-cleq 2405  df-clel 2408  df-nfc 2537  df-ne 2577  df-rab 2683  df-v 2926  df-dif 3291  df-un 3293  df-in 3295  df-ss 3302  df-nul 3597  df-if 3708  df-sn 3788  df-pr 3789  df-op 3791  df-opab 4235  df-oprab 6052
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