Theorem ralrimdva

Description: Inference from Theorem 19.21 of Margaris p. 90. (Restricted quantifier version.) (Contributed by NM, 2-Feb-2008) (Proof shortened by Wolf Lammen, 28-Dec-2019)

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

Proof

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