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Theorem cbvralv2 2969
Description: Rule used to change the bound variable in a restricted universal quantifier with implicit substitution which also changes the quantifier domain. (Contributed by David Moews, 1-May-2017.)
Hypotheses
Ref Expression
cbvralv2.1  |-  ( x  =  y  ->  ( ps 
<->  ch ) )
cbvralv2.2  |-  ( x  =  y  ->  A  =  B )
Assertion
Ref Expression
cbvralv2  |-  ( A. x  e.  A  ps  <->  A. y  e.  B  ch )
Distinct variable groups:    y, A    ps, y    x, B    ch, x
Allowed substitution hints:    ps( x)    ch( y)    A( x)    B( y)

Proof of Theorem cbvralv2
StepHypRef Expression
1 nfcv 2220 . 2  |-  F/_ y A
2 nfcv 2220 . 2  |-  F/_ x B
3 nfv 1462 . 2  |-  F/ y ps
4 nfv 1462 . 2  |-  F/ x ch
5 cbvralv2.2 . 2  |-  ( x  =  y  ->  A  =  B )
6 cbvralv2.1 . 2  |-  ( x  =  y  ->  ( ps 
<->  ch ) )
71, 2, 3, 4, 5, 6cbvralcsf 2965 1  |-  ( A. x  e.  A  ps  <->  A. y  e.  B  ch )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 103    = wceq 1285   A.wral 2349
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2064
This theorem depends on definitions:  df-bi 115  df-tru 1288  df-nf 1391  df-sb 1687  df-clab 2069  df-cleq 2075  df-clel 2078  df-nfc 2209  df-ral 2354  df-sbc 2817  df-csb 2910
This theorem is referenced by: (None)
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