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Theorem cbvoprab3v 6273
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 1674 . 2  |-  F/ w ph
2 nfv 1674 . 2  |-  F/ z ps
3 cbvoprab3v.1 . 2  |-  ( z  =  w  ->  ( ph 
<->  ps ) )
41, 2, 3cbvoprab3 6272 1  |-  { <. <.
x ,  y >. ,  z >.  |  ph }  =  { <. <. x ,  y >. ,  w >.  |  ps }
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    = wceq 1370   {coprab 6202
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432  ax-sep 4522  ax-nul 4530  ax-pr 4640
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2650  df-rab 2808  df-v 3080  df-dif 3440  df-un 3442  df-in 3444  df-ss 3451  df-nul 3747  df-if 3901  df-sn 3987  df-pr 3989  df-op 3993  df-opab 4460  df-oprab 6205
This theorem is referenced by: (None)
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