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Theorem rexbida 2492
Description: Formula-building rule for restricted existential quantifier (deduction form). (Contributed by NM, 6-Oct-2003.)
Hypotheses
Ref Expression
ralbida.1  |-  F/ x ph
ralbida.2  |-  ( (
ph  /\  x  e.  A )  ->  ( ps 
<->  ch ) )
Assertion
Ref Expression
rexbida  |-  ( ph  ->  ( E. x  e.  A  ps  <->  E. x  e.  A  ch )
)

Proof of Theorem rexbida
StepHypRef Expression
1 ralbida.1 . . 3  |-  F/ x ph
2 ralbida.2 . . . 4  |-  ( (
ph  /\  x  e.  A )  ->  ( ps 
<->  ch ) )
32pm5.32da 452 . . 3  |-  ( ph  ->  ( ( x  e.  A  /\  ps )  <->  ( x  e.  A  /\  ch ) ) )
41, 3exbid 1630 . 2  |-  ( ph  ->  ( E. x ( x  e.  A  /\  ps )  <->  E. x ( x  e.  A  /\  ch ) ) )
5 df-rex 2481 . 2  |-  ( E. x  e.  A  ps  <->  E. x ( x  e.  A  /\  ps )
)
6 df-rex 2481 . 2  |-  ( E. x  e.  A  ch  <->  E. x ( x  e.  A  /\  ch )
)
74, 5, 63bitr4g 223 1  |-  ( ph  ->  ( E. x  e.  A  ps  <->  E. x  e.  A  ch )
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 104    <-> wb 105   F/wnf 1474   E.wex 1506    e. wcel 2167   E.wrex 2476
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-5 1461  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-4 1524  ax-ial 1548
This theorem depends on definitions:  df-bi 117  df-nf 1475  df-rex 2481
This theorem is referenced by:  rexbidva  2494  rexbid  2496  rexbi  2630  dfiun2g  3948  fun11iun  5525  ismkvnex  7221  mkvprop  7224
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