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Theorem cbvopab1v 4092
Description: Rule used to change the first bound variable in an ordered pair abstraction, using implicit substitution. (Contributed by NM, 31-Jul-2003.) (Proof shortened by Eric Schmidt, 4-Apr-2007.)
Hypothesis
Ref Expression
cbvopab1v.1  |-  ( x  =  z  ->  ( ph 
<->  ps ) )
Assertion
Ref Expression
cbvopab1v  |-  { <. x ,  y >.  |  ph }  =  { <. z ,  y >.  |  ps }
Distinct variable groups:    x, y    y, z    ph, z    ps, x
Allowed substitution hints:    ph( x, y)    ps( y, z)

Proof of Theorem cbvopab1v
StepHypRef Expression
1 nfv 1605 . 2  |-  F/ z
ph
2 nfv 1605 . 2  |-  F/ x ps
3 cbvopab1v.1 . 2  |-  ( x  =  z  ->  ( ph 
<->  ps ) )
41, 2, 3cbvopab1 4089 1  |-  { <. x ,  y >.  |  ph }  =  { <. z ,  y >.  |  ps }
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    = wceq 1623   {copab 4076
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-rab 2552  df-v 2790  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-opab 4078
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