Theorem rexlimd3

Description: * Inference from Theorem 19.23 of Margaris p. 90 (restricted quantifier version). (Contributed by Glauco Siliprandi, 2-Jan-2022)

	Ref	Expression
Hypotheses	rexlimd3.1	$$⊢Ⅎx\phi$$
	rexlimd3.2	$$⊢Ⅎx\chi$$
	rexlimd3.3	$$⊢\left(\left(\phi \wedge x\in A\right)\wedge \psi \right)\to \chi$$
Assertion	rexlimd3	$$⊢\phi \to \left(\exists x\in A\psi \to \chi \right)$$

Proof

Step	Hyp	Ref	Expression
1		rexlimd3.1	$$⊢Ⅎx\phi$$
2		rexlimd3.2	$$⊢Ⅎx\chi$$
3		rexlimd3.3	$$⊢\left(\left(\phi \wedge x\in A\right)\wedge \psi \right)\to \chi$$
4	3	exp31	$$⊢\phi \to \left(x\in A\to \left(\psi \to \chi \right)\right)$$
5	1 2 4	rexlimd	$$⊢\phi \to \left(\exists x\in A\psi \to \chi \right)$$