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Theorem rexeqbid 3036
Description: Equality deduction for restricted existential quantifier. (Contributed by Thierry Arnoux, 8-Mar-2017.)
Hypotheses
Ref Expression
raleqbid.0  |-  F/ x ph
raleqbid.1  |-  F/_ x A
raleqbid.2  |-  F/_ x B
raleqbid.3  |-  ( ph  ->  A  =  B )
raleqbid.4  |-  ( ph  ->  ( ps  <->  ch )
)
Assertion
Ref Expression
rexeqbid  |-  ( ph  ->  ( E. x  e.  A  ps  <->  E. x  e.  B  ch )
)

Proof of Theorem rexeqbid
StepHypRef Expression
1 raleqbid.3 . . 3  |-  ( ph  ->  A  =  B )
2 raleqbid.1 . . . 4  |-  F/_ x A
3 raleqbid.2 . . . 4  |-  F/_ x B
42, 3rexeqf 3020 . . 3  |-  ( A  =  B  ->  ( E. x  e.  A  ps 
<->  E. x  e.  B  ps ) )
51, 4syl 17 . 2  |-  ( ph  ->  ( E. x  e.  A  ps  <->  E. x  e.  B  ps )
)
6 raleqbid.0 . . 3  |-  F/ x ph
7 raleqbid.4 . . 3  |-  ( ph  ->  ( ps  <->  ch )
)
86, 7rexbid 2936 . 2  |-  ( ph  ->  ( E. x  e.  B  ps  <->  E. x  e.  B  ch )
)
95, 8bitrd 256 1  |-  ( ph  ->  ( E. x  e.  A  ps  <->  E. x  e.  B  ch )
)
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 187    = wceq 1437   F/wnf 1663   F/_wnfc 2568   E.wrex 2774
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1748  ax-6 1794  ax-7 1838  ax-10 1886  ax-11 1891  ax-12 1904  ax-ext 2398
This theorem depends on definitions:  df-bi 188  df-an 372  df-tru 1440  df-ex 1660  df-nf 1664  df-cleq 2412  df-clel 2415  df-nfc 2570  df-rex 2779
This theorem is referenced by:  iuneq12df  4317
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