Theorem exlimdd 1865

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 114	. . 3 |- ( ph -> ( ps -> ch ) )
6	2, 3, 5	exlimd 1590	. 2 |- ( ph -> ( E. x ps -> ch ) )
7	1, 6	mpd 13	1 |- ( ph -> ch )

Syntax hints:   -> wi 4   /\ wa 103   F/wnf 1453   E.wex 1485

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia3 107  ax-5 1440  ax-gen 1442  ax-ie2 1487  ax-4 1503

This theorem depends on definitions:  df-bi 116  df-nf 1454

This theorem is referenced by:  fvmptdf  5583  ovmpodf  5984  exmidfodomrlemr  7179  exmidfodomrlemrALT  7180  ltexprlemm  7562