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Theorem exlimdd 1842
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 423 . . 3  |-  ( ph  ->  ( ps  ->  ch ) )
62, 3, 5exlimd 1815 . 2  |-  ( ph  ->  ( E. x ps 
->  ch ) )
71, 6mpd 14 1  |-  ( ph  ->  ch )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358   E.wex 1531   F/wnf 1534
This theorem is referenced by:  ex-natded9.26  20822
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
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