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Theorem rexlimd 2578
Description: Deduction from Theorem 19.23 of [Margaris] p. 90 (restricted quantifier version). (Contributed by NM, 27-May-1998.) (Proof shortened by Andrew Salmon, 30-May-2011.)
Hypotheses
Ref Expression
rexlimd.1  |-  F/ x ph
rexlimd.2  |-  F/ x ch
rexlimd.3  |-  ( ph  ->  ( x  e.  A  ->  ( ps  ->  ch ) ) )
Assertion
Ref Expression
rexlimd  |-  ( ph  ->  ( E. x  e.  A  ps  ->  ch ) )

Proof of Theorem rexlimd
StepHypRef Expression
1 rexlimd.1 . . 3  |-  F/ x ph
2 rexlimd.3 . . 3  |-  ( ph  ->  ( x  e.  A  ->  ( ps  ->  ch ) ) )
31, 2ralrimi 2535 . 2  |-  ( ph  ->  A. x  e.  A  ( ps  ->  ch )
)
4 rexlimd.2 . . 3  |-  F/ x ch
54r19.23 2572 . 2  |-  ( A. x  e.  A  ( ps  ->  ch )  <->  ( E. x  e.  A  ps  ->  ch ) )
63, 5sylib 121 1  |-  ( ph  ->  ( E. x  e.  A  ps  ->  ch ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4   F/wnf 1447    e. wcel 2135   A.wral 2442   E.wrex 2443
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 1434  ax-gen 1436  ax-ie1 1480  ax-ie2 1481  ax-4 1497  ax-ial 1521  ax-i5r 1522
This theorem depends on definitions:  df-bi 116  df-nf 1448  df-ral 2447  df-rex 2448
This theorem is referenced by:  rexlimdv  2580  ralxfrALT  4442  fvmptt  5574  ffnfv  5640  nneneq  6817  ac6sfi  6858  prarloclem3step  7431  prmuloc2  7502  caucvgprprlemaddq  7643  axpre-suploclemres  7836  lbzbi  9548  divalglemeunn  11852  divalglemeuneg  11854  oddpwdclemdvds  12096  oddpwdclemndvds  12097  trirec0  13816
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