Theorem ralimdv

Description: Deduction quantifying both antecedent and consequent, based on Theorem 19.20 of Margaris p. 90 ( alim ). (Contributed by NM, 8-Oct-2003)

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

Proof

Step	Hyp	Ref	Expression
1		ralimdv.1	$$⊢\phi \to \left(\psi \to \chi \right)$$
2	1	adantr	$$⊢\left(\phi \wedge x\in A\right)\to \left(\psi \to \chi \right)$$
3	2	ralimdva	$$⊢\phi \to \left(\forall x\in A\psi \to \forall x\in A\chi \right)$$