Theorem exlimddv 1638

Description: Existential elimination rule of natural deduction. (Contributed by Mario Carneiro, 15-Jun-2016.)

Hypotheses

Ref	Expression
exlimddv.1	|- (ph -> E.xps)
exlimddv.2	|- ((ph /\ ps) -> ch)

Assertion

Ref	Expression
exlimddv	|- (ph -> ch)

Distinct variable groups:   ch,x   ph,x

Allowed substitution hint:   ps(x)

Proof of Theorem exlimddv

Step	Hyp	Ref	Expression
1		exlimddv.1	. 2 |- (ph -> E.xps)
2		exlimddv.2	. . . 4 |- ((ph /\ ps) -> ch)
3	2	ex 423	. . 3 |- (ph -> (ps -> ch))
4	3	exlimdv 1636	. 2 |- (ph -> (E.xps -> ch))
5	1, 4	mpd 14	1 |- (ph -> ch)

Syntax hints:   -> wi 4   /\ wa 358  E.wex 1541

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616

This theorem depends on definitions:  df-bi 177  df-an 360  df-ex 1542

This theorem is referenced by: (None)