Theorem ralrimd

Description: Inference from Theorem 19.21 of Margaris p. 90. (Restricted quantifier version.) For a version based on fewer axioms see ralrimdv . (Contributed by NM, 16-Feb-2004)

	Ref	Expression
Hypotheses	ralrimd.1	|- F/ x ph
	ralrimd.2	|- F/ x ps
	ralrimd.3	|- ( ph -> ( ps -> ( x e. A -> ch ) ) )
Assertion	ralrimd	|- ( ph -> ( ps -> A. x e. A ch ) )

Proof

Step	Hyp	Ref	Expression
1		ralrimd.1	|- F/ x ph
2		ralrimd.2	|- F/ x ps
3		ralrimd.3	|- ( ph -> ( ps -> ( x e. A -> ch ) ) )
4	1 2 3	alrimd	|- ( ph -> ( ps -> A. x ( x e. A -> ch ) ) )
5		df-ral	|- ( A. x e. A ch <-> A. x ( x e. A -> ch ) )
6	4 5	imbitrrdi	|- ( ph -> ( ps -> A. x e. A ch ) )