Theorem exlimdd 1925

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 115	. . 3 |- ( ph -> ( ps -> ch ) )
6	2, 3, 5	exlimd 1650	. 2 |- ( ph -> ( E. x ps -> ch ) )
7	1, 6	mpd 13	1 |- ( ph -> ch )

This proof depends on syntax axioms:   -> wi 4   /\ wa 104   F/wnf 1513   E.wex 1545

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia3 108  ax-5 1500  ax-gen 1502  ax-ie2 1547  ax-4 1563

This proof depends on definitions:  df-bi 117  df-nf 1514

This theorem is used by:  fvmptdf  5793  ovmpodf  6220  exmidfodomrlemr  7554  exmidfodomrlemrALT  7555  ltexprlemm  7967  dfgrp3mlem  13903