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Theorem reximdai 2664
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 2637 . 2  |-  ( ph  ->  A. x  e.  A  ( ps  ->  ch )
)
4 rexim 2660 . 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 1534    e. wcel 1696   A.wral 2556   E.wrex 2557
This theorem is referenced by:  reximdvai  2666  tz7.49  6473  hsmexlem2  8069  indexdom  26516  filbcmb  26535  infrglb  27825  climinf  27835  stoweidlem29  27881  stoweidlem31  27883  stoweidlem34  27886  stoweidlem35  27887  2reurex  28062  cdlemefr29exN  31213
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-11 1727
This theorem depends on definitions:  df-bi 177  df-an 360  df-ex 1532  df-nf 1535  df-ral 2561  df-rex 2562
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