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Theorem rexlimd2 2771
Description: Version of rexlimd 2770 with deduction version of second hypothesis. (Contributed by NM, 21-Jul-2013.) (Revised by Mario Carneiro, 8-Oct-2016.)
Hypotheses
Ref Expression
rexlimd2.1  |-  F/ x ph
rexlimd2.2  |-  ( ph  ->  F/ x ch )
rexlimd2.3  |-  ( ph  ->  ( x  e.  A  ->  ( ps  ->  ch ) ) )
Assertion
Ref Expression
rexlimd2  |-  ( ph  ->  ( E. x  e.  A  ps  ->  ch ) )

Proof of Theorem rexlimd2
StepHypRef Expression
1 rexlimd2.1 . . 3  |-  F/ x ph
2 rexlimd2.3 . . 3  |-  ( ph  ->  ( x  e.  A  ->  ( ps  ->  ch ) ) )
31, 2ralrimi 2730 . 2  |-  ( ph  ->  A. x  e.  A  ( ps  ->  ch )
)
4 rexlimd2.2 . . 3  |-  ( ph  ->  F/ x ch )
5 r19.23t 2763 . . 3  |-  ( F/ x ch  ->  ( A. x  e.  A  ( ps  ->  ch )  <->  ( E. x  e.  A  ps  ->  ch ) ) )
64, 5syl 16 . 2  |-  ( ph  ->  ( A. x  e.  A  ( ps  ->  ch )  <->  ( E. x  e.  A  ps  ->  ch ) ) )
73, 6mpbid 202 1  |-  ( ph  ->  ( E. x  e.  A  ps  ->  ch ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177   F/wnf 1550    e. wcel 1717   A.wral 2649   E.wrex 2650
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 1661  ax-8 1682  ax-6 1736  ax-11 1753
This theorem depends on definitions:  df-bi 178  df-an 361  df-ex 1548  df-nf 1551  df-ral 2654  df-rex 2655
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