Theorem bj-cbveximi

Description: An equality-free general instance of one half of a precise form of bj-cbvex . (Contributed by BJ, 12-Mar-2023) (Proof modification is discouraged.)

	Ref	Expression
Hypotheses	bj-cbvalimi.maj	$$⊢\chi \to \left(\phi \to \psi \right)$$
	bj-cbveximi.denote	$$⊢\forall x\exists y\chi$$
Assertion	bj-cbveximi	$$⊢\exists x\phi \to \exists y\psi$$

Proof

Step	Hyp	Ref	Expression
1		bj-cbvalimi.maj	$$⊢\chi \to \left(\phi \to \psi \right)$$
2		bj-cbveximi.denote	$$⊢\forall x\exists y\chi$$
3	1	gen2	$$⊢\forall x\forall y\left(\chi \to \left(\phi \to \psi \right)\right)$$
4		bj-cbvexim	$$⊢\forall x\exists y\chi \to \left(\forall x\forall y\left(\chi \to \left(\phi \to \psi \right)\right)\to \left(\exists x\phi \to \exists y\psi \right)\right)$$
5	2 3 4	mp2	$$⊢\exists x\phi \to \exists y\psi$$