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Theorem rexbida 2931
Description: Formula-building rule for restricted existential quantifier (deduction rule). (Contributed by NM, 6-Oct-2003.)
Hypotheses
Ref Expression
rexbida.1  |-  F/ x ph
rexbida.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 rexbida.1 . . 3  |-  F/ x ph
2 rexbida.2 . . . 4  |-  ( (
ph  /\  x  e.  A )  ->  ( ps 
<->  ch ) )
32pm5.32da 645 . . 3  |-  ( ph  ->  ( ( x  e.  A  /\  ps )  <->  ( x  e.  A  /\  ch ) ) )
41, 3exbid 1941 . 2  |-  ( ph  ->  ( E. x ( x  e.  A  /\  ps )  <->  E. x ( x  e.  A  /\  ch ) ) )
5 df-rex 2777 . 2  |-  ( E. x  e.  A  ps  <->  E. x ( x  e.  A  /\  ps )
)
6 df-rex 2777 . 2  |-  ( E. x  e.  A  ch  <->  E. x ( x  e.  A  /\  ch )
)
74, 5, 63bitr4g 291 1  |-  ( ph  ->  ( E. x  e.  A  ps  <->  E. x  e.  A  ch )
)
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 187    /\ wa 370   E.wex 1657   F/wnf 1661    e. wcel 1872   E.wrex 2772
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1663  ax-4 1676  ax-5 1752  ax-6 1798  ax-7 1843  ax-12 1909
This theorem depends on definitions:  df-bi 188  df-an 372  df-ex 1658  df-nf 1662  df-rex 2777
This theorem is referenced by:  rexbidvaALT  2934  rexbid  2935  dfiun2g  4331  fun11iun  6767  iuneq12daf  28172  bnj1366  29649  glbconxN  32912
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