Theorem exlimdd 1749

Description: Existential elimination rule of natural deduction. (Contributed by Mario Carneiro, 9-Feb-2017.)

Hypotheses

Ref	Expression
exlimdd.1	|- F/xph
exlimdd.2	|- F/xch
exlimdd.3	|- (ph -> E.xps)
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.xps)
2		exlimdd.1	. . 3 |- F/xph
3		exlimdd.2	. . 3 |- F/xch
4		exlimdd.4	. . . 4 |- ((ph /\ ps) -> ch)
5	4	ex 108	. . 3 |- (ph -> (ps -> ch))
6	2, 3, 5	exlimd 1485	. 2 |- (ph -> (E.xps -> ch))
7	1, 6	mpd 13	1 |- (ph -> ch)

Syntax hints:   -> wi 4   /\ wa 97   F/wnf 1346  E.wex 1378

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia3 101  ax-5 1333  ax-gen 1335  ax-ie2 1380  ax-4 1397

This theorem depends on definitions:  df-bi 110  df-nf 1347

This theorem is referenced by:  fvmptdf  5201  ovmpt2df  5574  ltexprlemm  6574