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Theorem cbvopab1v 4055
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 1515 . 2  |-  F/ z
ph
2 nfv 1515 . 2  |-  F/ x ps
3 cbvopab1v.1 . 2  |-  ( x  =  z  ->  ( ph 
<->  ps ) )
41, 2, 3cbvopab1 4052 1  |-  { <. x ,  y >.  |  ph }  =  { <. z ,  y >.  |  ps }
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 104    = wceq 1342   {copab 4039
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1434  ax-7 1435  ax-gen 1436  ax-ie1 1480  ax-ie2 1481  ax-8 1491  ax-10 1492  ax-11 1493  ax-i12 1494  ax-bndl 1496  ax-4 1497  ax-17 1513  ax-i9 1517  ax-ial 1521  ax-i5r 1522  ax-ext 2146
This theorem depends on definitions:  df-bi 116  df-3an 969  df-tru 1345  df-nf 1448  df-sb 1750  df-clab 2151  df-cleq 2157  df-clel 2160  df-nfc 2295  df-v 2726  df-un 3118  df-sn 3579  df-pr 3580  df-op 3582  df-opab 4041
This theorem is referenced by: (None)
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