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Theorem rexeqbid 2986
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 2970 . . 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 2891 . 2  |-  ( ph  ->  ( E. x  e.  B  ps  <->  E. x  e.  B  ch )
)
95, 8bitrd 261 1  |-  ( ph  ->  ( E. x  e.  A  ps  <->  E. x  e.  B  ch )
)
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 189    = wceq 1452   F/wnf 1675   F/_wnfc 2599   E.wrex 2757
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-10 1932  ax-11 1937  ax-12 1950  ax-ext 2451
This theorem depends on definitions:  df-bi 190  df-an 378  df-tru 1455  df-ex 1672  df-nf 1676  df-cleq 2464  df-clel 2467  df-nfc 2601  df-rex 2762
This theorem is referenced by:  iuneq12df  4293
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