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Theorem rexbida 2571
Description: Formula-building rule for restricted existential quantifier (deduction rule). (Contributed by NM, 6-Oct-2003.)
Hypotheses
Ref Expression
ralbida.1  |-  F/ x ph
ralbida.2  |-  ( (
ph  /\  x  e.  A )  ->  ( ps 
<->  ch ) )
Assertion
Ref Expression
rexbida  |-  ( ph  ->  ( E. x  e.  A  ps  <->  E. x  e.  A  ch )
)

Proof of Theorem rexbida
StepHypRef Expression
1 ralbida.1 . . 3  |-  F/ x ph
2 ralbida.2 . . . 4  |-  ( (
ph  /\  x  e.  A )  ->  ( ps 
<->  ch ) )
32pm5.32da 622 . . 3  |-  ( ph  ->  ( ( x  e.  A  /\  ps )  <->  ( x  e.  A  /\  ch ) ) )
41, 3exbid 1765 . 2  |-  ( ph  ->  ( E. x ( x  e.  A  /\  ps )  <->  E. x ( x  e.  A  /\  ch ) ) )
5 df-rex 2562 . 2  |-  ( E. x  e.  A  ps  <->  E. x ( x  e.  A  /\  ps )
)
6 df-rex 2562 . 2  |-  ( E. x  e.  A  ch  <->  E. x ( x  e.  A  /\  ch )
)
74, 5, 63bitr4g 279 1  |-  ( ph  ->  ( E. x  e.  A  ps  <->  E. x  e.  A  ch )
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358   E.wex 1531   F/wnf 1534    e. wcel 1696   E.wrex 2557
This theorem is referenced by:  rexbidva  2573  rexbid  2575  dfiun2g  3951  fun11iun  5509  ballotlemsima  23090  iuneq12daf  23170  bnj1366  29178  glbconxN  30189
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-11 1727
This theorem depends on definitions:  df-bi 177  df-an 360  df-ex 1532  df-nf 1535  df-rex 2562
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