Theorem bj-exalims

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

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

Proof

Step	Hyp	Ref	Expression
1		bj-exalims.1	$$⊢\exists x\phi \to \left(\neg \chi \to \forall x\neg \chi \right)$$
2		bj-exalim	$$⊢\forall x\left(\phi \to \left(\psi \to \chi \right)\right)\to \left(\exists x\phi \to \left(\forall x\psi \to \exists x\chi \right)\right)$$
3		eximal	$$⊢\left(\exists x\chi \to \chi \right)\leftrightarrow \left(\neg \chi \to \forall x\neg \chi \right)$$
4	1 3	sylibr	$$⊢\exists x\phi \to \left(\exists x\chi \to \chi \right)$$
5	4	a1i	$$⊢\forall x\left(\phi \to \left(\psi \to \chi \right)\right)\to \left(\exists x\phi \to \left(\exists x\chi \to \chi \right)\right)$$
6	2 5	syldd	$$⊢\forall x\left(\phi \to \left(\psi \to \chi \right)\right)\to \left(\exists x\phi \to \left(\forall x\psi \to \chi \right)\right)$$