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Theorem rexbid2 2469
Description: Formula-building rule for restricted existential quantifier (deduction form). (Contributed by BJ, 14-Jul-2024.)
Hypotheses
Ref Expression
rexbid2.nf  |-  F/ x ph
rexbid2.1  |-  ( ph  ->  ( ( x  e.  A  /\  ps )  <->  ( x  e.  B  /\  ch ) ) )
Assertion
Ref Expression
rexbid2  |-  ( ph  ->  ( E. x  e.  A  ps  <->  E. x  e.  B  ch )
)

Proof of Theorem rexbid2
StepHypRef Expression
1 rexbid2.nf . . 3  |-  F/ x ph
2 rexbid2.1 . . 3  |-  ( ph  ->  ( ( x  e.  A  /\  ps )  <->  ( x  e.  B  /\  ch ) ) )
31, 2exbid 1603 . 2  |-  ( ph  ->  ( E. x ( x  e.  A  /\  ps )  <->  E. x ( x  e.  B  /\  ch ) ) )
4 df-rex 2448 . 2  |-  ( E. x  e.  A  ps  <->  E. x ( x  e.  A  /\  ps )
)
5 df-rex 2448 . 2  |-  ( E. x  e.  B  ch  <->  E. x ( x  e.  B  /\  ch )
)
63, 4, 53bitr4g 222 1  |-  ( ph  ->  ( E. x  e.  A  ps  <->  E. x  e.  B  ch )
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    <-> wb 104   F/wnf 1447   E.wex 1479    e. wcel 2135   E.wrex 2443
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1434  ax-gen 1436  ax-ie1 1480  ax-ie2 1481  ax-4 1497  ax-ial 1521
This theorem depends on definitions:  df-bi 116  df-nf 1448  df-rex 2448
This theorem is referenced by: (None)
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