MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  rexlimd2 Unicode version

Theorem rexlimd2 2678
Description: Version of rexlimd 2677 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 2637 . 2  |-  ( ph  ->  A. x  e.  A  ( ps  ->  ch )
)
4 rexlimd2.2 . . 3  |-  ( ph  ->  F/ x ch )
5 r19.23t 2670 . . 3  |-  ( F/ x ch  ->  ( A. x  e.  A  ( ps  ->  ch )  <->  ( E. x  e.  A  ps  ->  ch ) ) )
64, 5syl 15 . 2  |-  ( ph  ->  ( A. x  e.  A  ( ps  ->  ch )  <->  ( E. x  e.  A  ps  ->  ch ) ) )
73, 6mpbid 201 1  |-  ( ph  ->  ( E. x  e.  A  ps  ->  ch ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176   F/wnf 1534    e. wcel 1696   A.wral 2556   E.wrex 2557
This theorem is referenced by:  cdlemk19x  31754  cdlemk11t  31757
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-6 1715  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
  Copyright terms: Public domain W3C validator