Theorem ralrimdv

Description: Inference from Theorem 19.21 of Margaris p. 90. (Restricted quantifier version.) (Contributed by NM, 27-May-1998) Reduce dependencies on axioms. (Revised by Wolf Lammen, 28-Dec-2019)

	Ref	Expression
Hypothesis	ralrimdv.1	$$⊢\phi \to \left(\psi \to \left(x\in A\to \chi \right)\right)$$
Assertion	ralrimdv	$$⊢\phi \to \left(\psi \to \forall x\in A\chi \right)$$

Proof

Step	Hyp	Ref	Expression
1		ralrimdv.1	$$⊢\phi \to \left(\psi \to \left(x\in A\to \chi \right)\right)$$
2	1	imp	$$⊢\left(\phi \wedge \psi \right)\to \left(x\in A\to \chi \right)$$
3	2	ralrimiv	$$⊢\left(\phi \wedge \psi \right)\to \forall x\in A\chi$$
4	3	ex	$$⊢\phi \to \left(\psi \to \forall x\in A\chi \right)$$