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Theorem rexbid 2325
Description: Formula-building rule for restricted existential quantifier (deduction rule). (Contributed by NM, 27-Jun-1998.)
Hypotheses
Ref Expression
ralbid.1  |-  F/ x ph
ralbid.2  |-  ( ph  ->  ( ps  <->  ch )
)
Assertion
Ref Expression
rexbid  |-  ( ph  ->  ( E. x  e.  A  ps  <->  E. x  e.  A  ch )
)

Proof of Theorem rexbid
StepHypRef Expression
1 ralbid.1 . 2  |-  F/ x ph
2 ralbid.2 . . 3  |-  ( ph  ->  ( ps  <->  ch )
)
32adantr 261 . 2  |-  ( (
ph  /\  x  e.  A )  ->  ( ps 
<->  ch ) )
41, 3rexbida 2321 1  |-  ( ph  ->  ( E. x  e.  A  ps  <->  E. x  e.  A  ch )
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 98   F/wnf 1349    e. wcel 1393   E.wrex 2307
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-5 1336  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-4 1400  ax-ial 1427
This theorem depends on definitions:  df-bi 110  df-nf 1350  df-rex 2312
This theorem is referenced by:  rexbidv  2327  sbcrext  2835  caucvgsrlemgt1  6879  sscoll2  10113
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