Theorem rexlimd2

Description: Version of rexlimd with deduction version of second hypothesis. (Contributed by NM, 21-Jul-2013) (Revised by Mario Carneiro, 8-Oct-2016)

	Ref	Expression
Hypotheses	rexlimd2.1	$$⊢Ⅎx\phi$$
	rexlimd2.2	$$⊢\phi \to Ⅎx\chi$$
	rexlimd2.3	$$⊢\phi \to \left(x\in A\to \left(\psi \to \chi \right)\right)$$
Assertion	rexlimd2	$$⊢\phi \to \left(\exists x\in A\psi \to \chi \right)$$

Proof

Step	Hyp	Ref	Expression
1		rexlimd2.1	$$⊢Ⅎx\phi$$
2		rexlimd2.2	$$⊢\phi \to Ⅎx\chi$$
3		rexlimd2.3	$$⊢\phi \to \left(x\in A\to \left(\psi \to \chi \right)\right)$$
4	1 3	ralrimi	$$⊢\phi \to \forall x\in A\left(\psi \to \chi \right)$$
5		r19.23t	$$⊢Ⅎx\chi \to \left(\forall x\in A\left(\psi \to \chi \right)\leftrightarrow \left(\exists x\in A\psi \to \chi \right)\right)$$
6	2 5	syl	$$⊢\phi \to \left(\forall x\in A\left(\psi \to \chi \right)\leftrightarrow \left(\exists x\in A\psi \to \chi \right)\right)$$
7	4 6	mpbid	$$⊢\phi \to \left(\exists x\in A\psi \to \chi \right)$$