Theorem exlimimd 33190

Description: Existential elimination rule of natural deduction. (Contributed by ML, 17-Jul-2020.)

Hypotheses

Ref	Expression
exlimimd.1	|- ( ph -> E. x ps )
exlimimd.2	|- ( ph -> ( ps -> ch ) )

Assertion

Ref	Expression
exlimimd	|- ( ph -> ch )

Distinct variable groups:   ph, x   ch, x

Allowed substitution hint:   ps( x)

Proof of Theorem exlimimd

Step	Hyp	Ref	Expression
1		exlimimd.1	. 2 |- ( ph -> E. x ps )
2		exlimimd.2	. . 3 |- ( ph -> ( ps -> ch ) )
3	2	imp 445	. 2 |- ( ( ph /\ ps ) -> ch )
4	1, 3	exlimddv 1863	1 |- ( ph -> ch )

Syntax hints:   -> wi 4   E.wex 1704

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

This theorem depends on definitions:  df-bi 197  df-an 386  df-ex 1705

This theorem is referenced by: (None)