Theorem bj-exalimsi

Description: An inference for distributing quantifiers over a nested implication. (Almost) the general statement that spimfw proves. (Contributed by BJ, 29-Sep-2019)

	Ref	Expression
Hypotheses	bj-exalimsi.1	$$⊢\phi \to \left(\psi \to \chi \right)$$
	bj-exalimsi.2	$$⊢\exists x\phi \to \left(\neg \chi \to \forall x\neg \chi \right)$$
Assertion	bj-exalimsi	$$⊢\exists x\phi \to \left(\forall x\psi \to \chi \right)$$

Proof

Step	Hyp	Ref	Expression
1		bj-exalimsi.1	$$⊢\phi \to \left(\psi \to \chi \right)$$
2		bj-exalimsi.2	$$⊢\exists x\phi \to \left(\neg \chi \to \forall x\neg \chi \right)$$
3	2	bj-exalims	$$⊢\forall x\left(\phi \to \left(\psi \to \chi \right)\right)\to \left(\exists x\phi \to \left(\forall x\psi \to \chi \right)\right)$$
4	3 1	mpg	$$⊢\exists x\phi \to \left(\forall x\psi \to \chi \right)$$