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Theorem exlimdd 1924
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 434 . . 3  |-  ( ph  ->  ( ps  ->  ch ) )
62, 3, 5exlimd 1856 . 2  |-  ( ph  ->  ( E. x ps 
->  ch ) )
71, 6mpd 15 1  |-  ( ph  ->  ch )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369   E.wex 1591   F/wnf 1594
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-10 1781  ax-12 1798
This theorem depends on definitions:  df-bi 185  df-an 371  df-ex 1592  df-nf 1595
This theorem is referenced by:  fvmptdf  5952  ovmpt2df  6409  ex-natded9.26  24803  stoweidlem43  31298  stoweidlem44  31299  stoweidlem54  31309
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