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Theorem cbvopabv 4104
Description: Rule used to change bound variables in an ordered-pair class abstraction, using implicit substitution. (Contributed by NM, 15-Oct-1996.)
Hypothesis
Ref Expression
cbvopabv.1  |-  ( ( x  =  z  /\  y  =  w )  ->  ( ph  <->  ps )
)
Assertion
Ref Expression
cbvopabv  |-  { <. x ,  y >.  |  ph }  =  { <. z ,  w >.  |  ps }
Distinct variable groups:    x, y,
z, w    ph, z, w    ps, x, y
Allowed substitution hints:    ph( x, y)    ps( z, w)

Proof of Theorem cbvopabv
StepHypRef Expression
1 nfv 1609 . 2  |-  F/ z
ph
2 nfv 1609 . 2  |-  F/ w ph
3 nfv 1609 . 2  |-  F/ x ps
4 nfv 1609 . 2  |-  F/ y ps
5 cbvopabv.1 . 2  |-  ( ( x  =  z  /\  y  =  w )  ->  ( ph  <->  ps )
)
61, 2, 3, 4, 5cbvopab 4103 1  |-  { <. x ,  y >.  |  ph }  =  { <. z ,  w >.  |  ps }
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    = wceq 1632   {copab 4092
This theorem is referenced by:  cantnf  7411  infxpen  7658  axdc2  8091  fpwwe2cbv  8268  fpwwecbv  8282  sylow1  14930  bcth  18767  vitali  18984  lgsquadlem3  20611  lgsquad  20612  cvmlift2lem13  23861  axcontlem1  24664  trnij  25718  bosser  26270  pellex  27023  aomclem8  27262
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-rab 2565  df-v 2803  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-sn 3659  df-pr 3660  df-op 3662  df-opab 4094
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