Theorem exlimdd

Description: Existential elimination rule of natural deduction. (Contributed by Mario Carneiro, 9-Feb-2017) (Proof shortened by Wolf Lammen, 3-Sep-2023)

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

Proof

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