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Theorem reximdai 2774
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 2747 . 2  |-  ( ph  ->  A. x  e.  A  ( ps  ->  ch )
)
4 rexim 2770 . 2  |-  ( A. x  e.  A  ( ps  ->  ch )  -> 
( E. x  e.  A  ps  ->  E. x  e.  A  ch )
)
53, 4syl 16 1  |-  ( ph  ->  ( E. x  e.  A  ps  ->  E. x  e.  A  ch )
)
Colors of variables: wff set class
Syntax hints:    -> wi 4   F/wnf 1550    e. wcel 1721   A.wral 2666   E.wrex 2667
This theorem is referenced by:  reximdvai  2776  tz7.49  6661  hsmexlem2  8263  indexdom  26326  filbcmb  26332  infrglb  27589  stoweidlem31  27647  stoweidlem34  27650  stoweidlem35  27651  2reurex  27826  cdlemefr29exN  30884
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-11 1757
This theorem depends on definitions:  df-bi 178  df-an 361  df-ex 1548  df-nf 1551  df-ral 2671  df-rex 2672
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