Theorem reximi2

Description: Inference quantifying both antecedent and consequent, based on Theorem 19.22 of Margaris p. 90. (Contributed by NM, 8-Nov-2004)

	Ref	Expression
Hypothesis	reximi2.1	$$⊢\left(x\in A\wedge \phi \right)\to \left(x\in B\wedge \psi \right)$$
Assertion	reximi2	$$⊢\exists x\in A\phi \to \exists x\in B\psi$$

Proof

Step	Hyp	Ref	Expression
1		reximi2.1	$$⊢\left(x\in A\wedge \phi \right)\to \left(x\in B\wedge \psi \right)$$
2	1	eximi	$$⊢\exists x\left(x\in A\wedge \phi \right)\to \exists x\left(x\in B\wedge \psi \right)$$
3		df-rex	$$⊢\exists x\in A\phi \leftrightarrow \exists x\left(x\in A\wedge \phi \right)$$
4		df-rex	$$⊢\exists x\in B\psi \leftrightarrow \exists x\left(x\in B\wedge \psi \right)$$
5	2 3 4	3imtr4i	$$⊢\exists x\in A\phi \to \exists x\in B\psi$$