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Metamath Proof Explorer

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	$⊢ χ \to (φ \to ψ)$
	bj-cbvalimi.denote	$⊢ \forall y \exists x χ$
Assertion	bj-cbvalimi	$⊢ \forall x φ \to \forall y ψ$

Proof

Step	Hyp	Ref	Expression
1		bj-cbvalimi.maj	$⊢ χ \to (φ \to ψ)$
2		bj-cbvalimi.denote	$⊢ \forall y \exists x χ$
3	1	gen2	$⊢ \forall y \forall x (χ \to (φ \to ψ))$
4		bj-cbvalim	$⊢ \forall y \exists x χ \to (\forall y \forall x (χ \to (φ \to ψ)) \to (\forall x φ \to \forall y ψ))$
5	2 3 4	mp2	$⊢ \forall x φ \to \forall y ψ$