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Theorem exlimdd 1912
Description: Existential elimination rule of natural deduction. (Contributed by Mario Carneiro, 9-Feb-2017.)
Hypotheses
Ref Expression
exlimdd.1  |-  F/ x ph
exlimdd.2  |-  F/ x ch
exlimdd.3  |-  ( ph  ->  E. x ps )
exlimdd.4  |-  ( (
ph  /\  ps )  ->  ch )
Assertion
Ref Expression
exlimdd  |-  ( ph  ->  ch )

Proof of Theorem exlimdd
StepHypRef Expression
1 exlimdd.3 . 2  |-  ( ph  ->  E. x ps )
2 exlimdd.1 . . 3  |-  F/ x ph
3 exlimdd.2 . . 3  |-  F/ x ch
4 exlimdd.4 . . . 4  |-  ( (
ph  /\  ps )  ->  ch )
54ex 424 . . 3  |-  ( ph  ->  ( ps  ->  ch ) )
62, 3, 5exlimd 1824 . 2  |-  ( ph  ->  ( E. x ps 
->  ch ) )
71, 6mpd 15 1  |-  ( ph  ->  ch )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359   E.wex 1550   F/wnf 1553
This theorem is referenced by:  fvmptdf  5807  ovmpt2df  6196  ex-natded9.26  21715  stoweidlem43  27706  stoweidlem44  27707  stoweidlem54  27717
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-6 1744  ax-11 1761
This theorem depends on definitions:  df-bi 178  df-an 361  df-ex 1551  df-nf 1554
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