Theorem exlimddvf 33926

Description: A lemma for eliminating an existential quantifier. (Contributed by Giovanni Mascellani, 30-May-2019.)

Hypotheses

Ref	Expression
exlimddvf.1	|- ( ph -> E. x th )
exlimddvf.2	|- F/ x ps
exlimddvf.3	|- ( ( th /\ ps ) -> ch )
exlimddvf.4	|- F/ x ch

Assertion

Ref	Expression
exlimddvf	|- ( ( ph /\ ps ) -> ch )

Proof of Theorem exlimddvf

Step	Hyp	Ref	Expression
1		exlimddvf.1	. 2 |- ( ph -> E. x th )
2		exlimddvf.2	. . 3 |- F/ x ps
3		exlimddvf.4	. . 3 |- F/ x ch
4		exlimddvf.3	. . . 4 |- ( ( th /\ ps ) -> ch )
5	4	expcom 451	. . 3 |- ( ps -> ( th -> ch ) )
6	2, 3, 5	exlimd 2087	. 2 |- ( ps -> ( E. x th -> ch ) )
7	1, 6	mpan9 486	1 |- ( ( ph /\ ps ) -> 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:  exlimddvfi  33927