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	$$⊢Ⅎx\phi$$
	ralrimd.2	$$⊢Ⅎx\psi$$
	ralrimd.3	$$⊢\phi \to \left(\psi \to \left(x\in A\to \chi \right)\right)$$
Assertion	ralrimd	$$⊢\phi \to \left(\psi \to \forall x\in A\chi \right)$$

Proof

Step	Hyp	Ref	Expression
1		ralrimd.1	$$⊢Ⅎx\phi$$
2		ralrimd.2	$$⊢Ⅎx\psi$$
3		ralrimd.3	$$⊢\phi \to \left(\psi \to \left(x\in A\to \chi \right)\right)$$
4	1 2 3	alrimd	$$⊢\phi \to \left(\psi \to \forall x\left(x\in A\to \chi \right)\right)$$
5		df-ral	$$⊢\forall x\in A\chi \leftrightarrow \forall x\left(x\in A\to \chi \right)$$
6	4 5	imbitrrdi	$$⊢\phi \to \left(\psi \to \forall x\in A\chi \right)$$