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Theorem rexbid 2489
Description: Formula-building rule for restricted existential quantifier (deduction form). (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 276 . 2  |-  ( (
ph  /\  x  e.  A )  ->  ( ps 
<->  ch ) )
41, 3rexbida 2485 1  |-  ( ph  ->  ( E. x  e.  A  ps  <->  E. x  e.  A  ch )
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 105   F/wnf 1471    e. wcel 2160   E.wrex 2469
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 1458  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-4 1521  ax-ial 1545
This theorem depends on definitions:  df-bi 117  df-nf 1472  df-rex 2474
This theorem is referenced by:  rexbidv  2491  sbcrext  3055  mkvprop  7185  caucvgsrlemgt1  7823  bezout  12043
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