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Theorem reubidva 2542
Description: Formula-building rule for restricted existential quantifier (deduction rule). (Contributed by NM, 13-Nov-2004.)
Hypothesis
Ref Expression
reubidva.1  |-  ( (
ph  /\  x  e.  A )  ->  ( ps 
<->  ch ) )
Assertion
Ref Expression
reubidva  |-  ( ph  ->  ( E! x  e.  A  ps  <->  E! x  e.  A  ch )
)
Distinct variable group:    ph, x
Allowed substitution hints:    ps( x)    ch( x)    A( x)

Proof of Theorem reubidva
StepHypRef Expression
1 nfv 1462 . 2  |-  F/ x ph
2 reubidva.1 . 2  |-  ( (
ph  /\  x  e.  A )  ->  ( ps 
<->  ch ) )
31, 2reubida 2541 1  |-  ( ph  ->  ( E! x  e.  A  ps  <->  E! x  e.  A  ch )
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 102    <-> wb 103    e. wcel 1434   E!wreu 2355
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 1377  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-4 1441  ax-17 1460  ax-ial 1468
This theorem depends on definitions:  df-bi 115  df-nf 1391  df-eu 1946  df-reu 2360
This theorem is referenced by:  reubidv  2543  f1ofveu  5579  srpospr  7231  icoshftf1o  9303  divalgb  10705
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