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Theorem reubidva 2723
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 1605 . 2  |-  F/ x ph
2 reubidva.1 . 2  |-  ( (
ph  /\  x  e.  A )  ->  ( ps 
<->  ch ) )
31, 2reubida 2722 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. wcel 1684   E!wreu 2545
This theorem is referenced by:  reubidv  2724  reuxfr2d  4557  reuxfrd  4559  exfo  5678  riotabidva  6321  f1ofveu  6339  zmax  10313  zbtwnre  10314  rebtwnz  10315  icoshftf1o  10759  divalgb  12603  1arith2  12975  ply1divalg2  19524  addinv  21019  pjhtheu2  21995  reuxfr3d  23138  reuxfr4d  23139  xrmulc1cn  23303  hdmap14lem14  32074
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-11 1715
This theorem depends on definitions:  df-bi 177  df-an 360  df-ex 1529  df-nf 1532  df-eu 2147  df-reu 2550
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