Theorem rexlimiva

Description: Inference from Theorem 19.23 of Margaris p. 90 (restricted quantifier version). (Contributed by NM, 18-Dec-2006) Shorten dependent theorems. (Revised by Wolf lammen, 23-Dec-2024)

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

Proof

Step	Hyp	Ref	Expression
1		rexlimiva.1	$$⊢\left(x\in A\wedge \phi \right)\to \psi$$
2		df-rex	$$⊢\exists x\in A\phi \leftrightarrow \exists x\left(x\in A\wedge \phi \right)$$
3	1	exlimiv	$$⊢\exists x\left(x\in A\wedge \phi \right)\to \psi$$
4	2 3	sylbi	$$⊢\exists x\in A\phi \to \psi$$