Theorem rexlimiv

Description: Inference from Theorem 19.23 of Margaris p. 90. (Restricted quantifier version.) (Contributed by NM, 20-Nov-1994) Reduce dependencies on axioms. (Revised by Wolf Lammen, 14-Jan-2020)

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

Proof

Step	Hyp	Ref	Expression
1		rexlimiv.1	$$⊢x\in A\to \left(\phi \to \psi \right)$$
2	1	rgen	$$⊢\forall x\in A\left(\phi \to \psi \right)$$
3		r19.23v	$$⊢\forall x\in A\left(\phi \to \psi \right)\leftrightarrow \left(\exists x\in A\phi \to \psi \right)$$
4	2 3	mpbi	$$⊢\exists x\in A\phi \to \psi$$