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Theorem rexbid 2379
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 270 . 2  |-  ( (
ph  /\  x  e.  A )  ->  ( ps 
<->  ch ) )
41, 3rexbida 2375 1  |-  ( ph  ->  ( E. x  e.  A  ps  <->  E. x  e.  A  ch )
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 103   F/wnf 1394    e. wcel 1438   E.wrex 2360
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-5 1381  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-4 1445  ax-ial 1472
This theorem depends on definitions:  df-bi 115  df-nf 1395  df-rex 2365
This theorem is referenced by:  rexbidv  2381  sbcrext  2914  caucvgsrlemgt1  7319  bezout  11082  sscoll2  11529
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