Theorem reximdvai

Description: Deduction quantifying both antecedent and consequent, based on Theorem 19.22 of Margaris p. 90. (Contributed by NM, 14-Nov-2002) Reduce dependencies on axioms. (Revised by Wolf Lammen, 8-Jan-2020)

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

Proof

Step	Hyp	Ref	Expression
1		reximdvai.1	$$⊢\phi \to \left(x\in A\to \left(\psi \to \chi \right)\right)$$
2	1	ralrimiv	$$⊢\phi \to \forall x\in A\left(\psi \to \chi \right)$$
3		rexim	$$⊢\forall x\in A\left(\psi \to \chi \right)\to \left(\exists x\in A\psi \to \exists x\in A\chi \right)$$
4	2 3	syl	$$⊢\phi \to \left(\exists x\in A\psi \to \exists x\in A\chi \right)$$