Theorem bj-exalims

Description: Distributing quantifiers over a nested implication. (Almost) the general statement that spimfw proves. (Contributed by BJ, 29-Sep-2019)

	Ref	Expression
Hypothesis	bj-exalims.1	|- ( E. x ph -> ( -. ch -> A. x -. ch ) )
Assertion	bj-exalims	|- ( A. x ( ph -> ( ps -> ch ) ) -> ( E. x ph -> ( A. x ps -> ch ) ) )

Proof

Step	Hyp	Ref	Expression
1		bj-exalims.1	|- ( E. x ph -> ( -. ch -> A. x -. ch ) )
2		bj-exalim	|- ( A. x ( ph -> ( ps -> ch ) ) -> ( E. x ph -> ( A. x ps -> E. x ch ) ) )
3		eximal	|- ( ( E. x ch -> ch ) <-> ( -. ch -> A. x -. ch ) )
4	1 3	sylibr	|- ( E. x ph -> ( E. x ch -> ch ) )
5	4	a1i	|- ( A. x ( ph -> ( ps -> ch ) ) -> ( E. x ph -> ( E. x ch -> ch ) ) )
6	2 5	syldd	|- ( A. x ( ph -> ( ps -> ch ) ) -> ( E. x ph -> ( A. x ps -> ch ) ) )