MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  cbvopabv Unicode version

Theorem cbvopabv 4088
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 1605 . 2  |-  F/ z
ph
2 nfv 1605 . 2  |-  F/ w ph
3 nfv 1605 . 2  |-  F/ x ps
4 nfv 1605 . 2  |-  F/ y ps
5 cbvopabv.1 . 2  |-  ( ( x  =  z  /\  y  =  w )  ->  ( ph  <->  ps )
)
61, 2, 3, 4, 5cbvopab 4087 1  |-  { <. x ,  y >.  |  ph }  =  { <. z ,  w >.  |  ps }
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    = wceq 1623   {copab 4076
This theorem is referenced by:  cantnf  7395  infxpen  7642  axdc2  8075  fpwwe2cbv  8252  fpwwecbv  8266  sylow1  14914  bcth  18751  vitali  18968  lgsquadlem3  20595  lgsquad  20596  cvmlift2lem13  23257  axcontlem1  24003  trnij  25027  bosser  25579  pellex  26332  aomclem8  26571
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
  Copyright terms: Public domain W3C validator