Theorem ralimda

Description: Deduction quantifying both antecedent and consequent. (Contributed by Glauco Siliprandi, 23-Oct-2021)

	Ref	Expression
Hypotheses	ralimda.1	|- F/ x ph
	ralimda.2	|- ( ( ph /\ x e. A ) -> ( ps -> ch ) )
Assertion	ralimda	|- ( ph -> ( A. x e. A ps -> A. x e. A ch ) )

Proof

Step	Hyp	Ref	Expression
1		ralimda.1	|- F/ x ph
2		ralimda.2	|- ( ( ph /\ x e. A ) -> ( ps -> ch ) )
3		nfra1	|- F/ x A. x e. A ps
4	1 3	nfan	|- F/ x ( ph /\ A. x e. A ps )
5		id	|- ( ( ph /\ x e. A ) -> ( ph /\ x e. A ) )
6	5	adantlr	|- ( ( ( ph /\ A. x e. A ps ) /\ x e. A ) -> ( ph /\ x e. A ) )
7		rspa	|- ( ( A. x e. A ps /\ x e. A ) -> ps )
8	7	adantll	|- ( ( ( ph /\ A. x e. A ps ) /\ x e. A ) -> ps )
9	6 8 2	sylc	|- ( ( ( ph /\ A. x e. A ps ) /\ x e. A ) -> ch )
10	4 9	ralrimia	|- ( ( ph /\ A. x e. A ps ) -> A. x e. A ch )
11	10	ex	|- ( ph -> ( A. x e. A ps -> A. x e. A ch ) )