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Theorem rexbida 2407
Description: Formula-building rule for restricted existential quantifier (deduction form). (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 445 . . 3  |-  ( ph  ->  ( ( x  e.  A  /\  ps )  <->  ( x  e.  A  /\  ch ) ) )
41, 3exbid 1578 . 2  |-  ( ph  ->  ( E. x ( x  e.  A  /\  ps )  <->  E. x ( x  e.  A  /\  ch ) ) )
5 df-rex 2397 . 2  |-  ( E. x  e.  A  ps  <->  E. x ( x  e.  A  /\  ps )
)
6 df-rex 2397 . 2  |-  ( E. x  e.  A  ch  <->  E. x ( x  e.  A  /\  ch )
)
74, 5, 63bitr4g 222 1  |-  ( ph  ->  ( E. x  e.  A  ps  <->  E. x  e.  A  ch )
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    <-> wb 104   F/wnf 1419   E.wex 1451    e. wcel 1463   E.wrex 2392
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1406  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-4 1470  ax-ial 1497
This theorem depends on definitions:  df-bi 116  df-nf 1420  df-rex 2397
This theorem is referenced by:  rexbidva  2409  rexbid  2411  rexbi  2540  dfiun2g  3813  fun11iun  5354  ismkvnex  6995  mkvprop  6998
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