Theorem exlimdd 2088

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

Step	Hyp	Ref	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 )
5	4	ex 450	. . 3 |- ( ph -> ( ps -> ch ) )
6	2, 3, 5	exlimd 2087	. 2 |- ( ph -> ( E. x ps -> ch ) )
7	1, 6	mpd 15	1 |- ( ph -> ch )

Syntax hints:   -> wi 4   /\ wa 384   E.wex 1704   F/wnf 1708

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-12 2047

This theorem depends on definitions:  df-bi 197  df-an 386  df-ex 1705  df-nf 1710

This theorem is referenced by:  fvmptd3f  6295  ovmpt2df  6792  ex-natded9.26  27276  exlimimdd  33191  suprnmpt  39355  stoweidlem43  40260  stoweidlem44  40261  stoweidlem54  40271