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Theorem rexlimd2 2915
Description: Version of rexlimd 2916 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 2832 . 2  |-  ( ph  ->  A. x  e.  A  ( ps  ->  ch )
)
4 rexlimd2.2 . . 3  |-  ( ph  ->  F/ x ch )
5 r19.23t 2910 . . 3  |-  ( F/ x ch  ->  ( A. x  e.  A  ( ps  ->  ch )  <->  ( E. x  e.  A  ps  ->  ch ) ) )
64, 5syl 17 . 2  |-  ( ph  ->  ( A. x  e.  A  ( ps  ->  ch )  <->  ( E. x  e.  A  ps  ->  ch ) ) )
73, 6mpbid 213 1  |-  ( ph  ->  ( E. x  e.  A  ps  ->  ch ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 187   F/wnf 1663    e. wcel 1870   A.wral 2782   E.wrex 2783
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1751  ax-6 1797  ax-7 1841  ax-10 1889  ax-12 1907
This theorem depends on definitions:  df-bi 188  df-an 372  df-ex 1660  df-nf 1664  df-ral 2787  df-rex 2788
This theorem is referenced by:  rexlimd  2916  sbcrext  3379
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