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Metamath Proof Explorer

Theorem bj-exalimsi

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

	Ref	Expression
Hypotheses	bj-exalimsi.1	$⊢ φ \to (ψ \to χ)$
	bj-exalimsi.2	$⊢ \exists x φ \to (\neg χ \to \forall x \neg χ)$
Assertion	bj-exalimsi	$⊢ \exists x φ \to (\forall x ψ \to χ)$

Proof

Step	Hyp	Ref	Expression
1		bj-exalimsi.1	$⊢ φ \to (ψ \to χ)$
2		bj-exalimsi.2	$⊢ \exists x φ \to (\neg χ \to \forall x \neg χ)$
3	2	bj-exalims	$⊢ \forall x (φ \to (ψ \to χ)) \to (\exists x φ \to (\forall x ψ \to χ))$
4	3 1	mpg	$⊢ \exists x φ \to (\forall x ψ \to χ)$