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Theorem cbvralv2 3301
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 2578 . 2  |-  F/_ y A
2 nfcv 2578 . 2  |-  F/_ x B
3 nfv 1630 . 2  |-  F/ y ps
4 nfv 1630 . 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 3297 1  |-  ( A. x  e.  A  ps  <->  A. y  e.  B  ch )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 178    = wceq 1653   A.wral 2711
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1668  ax-8 1689  ax-6 1746  ax-7 1751  ax-11 1763  ax-12 1953  ax-ext 2423
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2567  df-ral 2716  df-sbc 3168  df-csb 3268
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