Theorem bj-cbvalimi

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

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

Proof

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