Theorem bj-exalimi

Description: An inference for distributing quantifiers over a nested implication. The canonical derivation from its closed form bj-exalim (using mpg ) has fewer essential steps, but more steps in total (yielding a longer compressed proof). (Almost) the general statement that speimfw proves. (Contributed by BJ, 29-Sep-2019)

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

Proof

Step	Hyp	Ref	Expression
1		bj-exalimi.1	$$⊢\phi \to \left(\psi \to \chi \right)$$
2	1	com12	$$⊢\psi \to \left(\phi \to \chi \right)$$
3	2	aleximi	$$⊢\forall x\psi \to \left(\exists x\phi \to \exists x\chi \right)$$
4	3	com12	$$⊢\exists x\phi \to \left(\forall x\psi \to \exists x\chi \right)$$