Theorem ralimdaa

Description: Deduction quantifying both antecedent and consequent, based on Theorem 19.20 of Margaris p. 90. (Contributed by NM, 22-Sep-2003) (Proof shortened by Wolf Lammen, 29-Dec-2019)

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

Proof

Step	Hyp	Ref	Expression
1		ralimdaa.1	|- F/ x ph
2		ralimdaa.2	|- ( ( ph /\ x e. A ) -> ( ps -> ch ) )
3	2	ex	|- ( ph -> ( x e. A -> ( ps -> ch ) ) )
4	1 3	ralrimi	|- ( ph -> A. x e. A ( ps -> ch ) )
5		ralim	|- ( A. x e. A ( ps -> ch ) -> ( A. x e. A ps -> A. x e. A ch ) )
6	4 5	syl	|- ( ph -> ( A. x e. A ps -> A. x e. A ch ) )