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Theorem reximdai 2653
Description: Deduction from Theorem 19.22 of [Margaris] p. 90. (Restricted quantifier version.) (Contributed by NM, 31-Aug-1999.)
Hypotheses
Ref Expression
reximdai.1  |-  F/ x ph
reximdai.2  |-  ( ph  ->  ( x  e.  A  ->  ( ps  ->  ch ) ) )
Assertion
Ref Expression
reximdai  |-  ( ph  ->  ( E. x  e.  A  ps  ->  E. x  e.  A  ch )
)

Proof of Theorem reximdai
StepHypRef Expression
1 reximdai.1 . . 3  |-  F/ x ph
2 reximdai.2 . . 3  |-  ( ph  ->  ( x  e.  A  ->  ( ps  ->  ch ) ) )
31, 2ralrimi 2626 . 2  |-  ( ph  ->  A. x  e.  A  ( ps  ->  ch )
)
4 rexim 2649 . 2  |-  ( A. x  e.  A  ( ps  ->  ch )  -> 
( E. x  e.  A  ps  ->  E. x  e.  A  ch )
)
53, 4syl 15 1  |-  ( ph  ->  ( E. x  e.  A  ps  ->  E. x  e.  A  ch )
)
Colors of variables: wff set class
Syntax hints:    -> wi 4   F/wnf 1533    e. wcel 1686   A.wral 2545   E.wrex 2546
This theorem is referenced by:  reximdvai  2655  tz7.49  6459  hsmexlem2  8055  indexdom  26424  filbcmb  26443  infrglb  27733  climinf  27743  stoweidlem29  27789  stoweidlem31  27791  stoweidlem34  27794  stoweidlem35  27795  2reurex  27970  cdlemefr29exN  30664
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1535  ax-5 1546  ax-17 1605  ax-9 1637  ax-8 1645  ax-11 1717
This theorem depends on definitions:  df-bi 177  df-an 360  df-ex 1531  df-nf 1534  df-ral 2550  df-rex 2551
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