Theorem exlimimdd 33191

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

Hypotheses

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

Assertion

Ref	Expression
exlimimdd	|- ( ph -> ch )

Proof of Theorem exlimimdd

Step	Hyp	Ref	Expression
1		exlimimdd.1	. 2 |- F/ x ph
2		exlimimdd.2	. 2 |- F/ x ch
3		exlimimdd.3	. 2 |- ( ph -> E. x ps )
4		exlimimdd.4	. . 3 |- ( ph -> ( ps -> ch ) )
5	4	imp 445	. 2 |- ( ( ph /\ ps ) -> ch )
6	1, 2, 3, 5	exlimdd 2088	1 |- ( ph -> ch )

Syntax hints:   -> wi 4   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: (None)