Theorem ralimdva

Description: Deduction quantifying both antecedent and consequent, based on Theorem 19.20 of Margaris p. 90. (Contributed by NM, 22-May-1999) Reduce dependencies on axioms. (Revised by Wolf Lammen, 5-Dec-2019)

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

Proof

Step	Hyp	Ref	Expression
1		ralimdva.1	$$⊢\left(\phi \wedge x\in A\right)\to \left(\psi \to \chi \right)$$
2	1	ex	$$⊢\phi \to \left(x\in A\to \left(\psi \to \chi \right)\right)$$
3	2	a2d	$$⊢\phi \to \left(\left(x\in A\to \psi \right)\to \left(x\in A\to \chi \right)\right)$$
4	3	ralimdv2	$$⊢\phi \to \left(\forall x\in A\psi \to \forall x\in A\chi \right)$$