Theorem reximdv2

Description: Deduction quantifying both antecedent and consequent, based on Theorem 19.22 of Margaris p. 90. (Contributed by NM, 17-Sep-2003)

	Ref	Expression
Hypothesis	reximdv2.1	$$⊢\phi \to \left(\left(x\in A\wedge \psi \right)\to \left(x\in B\wedge \chi \right)\right)$$
Assertion	reximdv2	$$⊢\phi \to \left(\exists x\in A\psi \to \exists x\in B\chi \right)$$

Proof

Step	Hyp	Ref	Expression
1		reximdv2.1	$$⊢\phi \to \left(\left(x\in A\wedge \psi \right)\to \left(x\in B\wedge \chi \right)\right)$$
2	1	eximdv	$$⊢\phi \to \left(\exists x\left(x\in A\wedge \psi \right)\to \exists x\left(x\in B\wedge \chi \right)\right)$$
3		df-rex	$$⊢\exists x\in A\psi \leftrightarrow \exists x\left(x\in A\wedge \psi \right)$$
4		df-rex	$$⊢\exists x\in B\chi \leftrightarrow \exists x\left(x\in B\wedge \chi \right)$$
5	2 3 4	3imtr4g	$$⊢\phi \to \left(\exists x\in A\psi \to \exists x\in B\chi \right)$$