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Theorem cbvrexv2 3280
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 2544 . 2  |-  F/_ y A
2 nfcv 2544 . 2  |-  F/_ x B
3 nfv 1626 . 2  |-  F/ y ps
4 nfv 1626 . 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 3276 1  |-  ( E. x  e.  A  ps  <->  E. y  e.  B  ch )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    = wceq 1649   E.wrex 2671
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2389
This theorem depends on definitions:  df-bi 178  df-an 361  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-clab 2395  df-cleq 2401  df-clel 2404  df-nfc 2533  df-ral 2675  df-rex 2676  df-sbc 3126  df-csb 3216
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