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Theorem rexbid 2867
Description: Formula-building rule for restricted existential quantifier (deduction rule). (Contributed by NM, 27-Jun-1998.)
Hypotheses
Ref Expression
rexbid.1  |-  F/ x ph
rexbid.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 rexbid.1 . 2  |-  F/ x ph
2 rexbid.2 . . 3  |-  ( ph  ->  ( ps  <->  ch )
)
32adantr 465 . 2  |-  ( (
ph  /\  x  e.  A )  ->  ( ps 
<->  ch ) )
41, 3rexbida 2864 1  |-  ( ph  ->  ( E. x  e.  A  ps  <->  E. x  e.  A  ch )
)
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184   F/wnf 1590    e. wcel 1758   E.wrex 2800
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-12 1794
This theorem depends on definitions:  df-bi 185  df-an 371  df-ex 1588  df-nf 1591  df-rex 2805
This theorem is referenced by:  rexbidvALT  2869  rexeqbid  3036  scott0  8207  infcvgaux1i  13440  bnj1463  32398
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