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Theorem cbvrexv2 2980
Description: Rule used to change the bound variable in a restricted existential 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
cbvrexv2  |-  ( E. x  e.  A  ps  <->  E. 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 cbvrexv2
StepHypRef Expression
1 nfcv 2223 . 2  |-  F/_ y A
2 nfcv 2223 . 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, 6cbvrexcsf 2976 1  |-  ( E. x  e.  A  ps  <->  E. y  e.  B  ch )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 103    = wceq 1285   E.wrex 2354
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 2065
This theorem depends on definitions:  df-bi 115  df-tru 1288  df-nf 1391  df-sb 1688  df-clab 2070  df-cleq 2076  df-clel 2079  df-nfc 2212  df-rex 2359  df-sbc 2827  df-csb 2920
This theorem is referenced by: (None)
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