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Metamath Proof Explorer

Theorem eximal

Description: An equivalence between an implication with an existentially quantified antecedent and an implication with a universally quantified consequent. An interesting case is when the same formula is substituted for both ph and ps , since then both implications express a type of nonfreeness. See also alimex . (Contributed by BJ, 12-May-2019)

		Ref	Expression
	Assertion	eximal	$⊢ (\exists x φ \to ψ) \leftrightarrow (\neg ψ \to \forall x \neg φ)$

Proof

Step	Hyp	Ref	Expression
1		df-ex	$⊢ \exists x φ \leftrightarrow \neg \forall x \neg φ$
2	1	imbi1i	$⊢ (\exists x φ \to ψ) \leftrightarrow (\neg \forall x \neg φ \to ψ)$
3		con1b	$⊢ (\neg \forall x \neg φ \to ψ) \leftrightarrow (\neg ψ \to \forall x \neg φ)$
4	2 3	bitri	$⊢ (\exists x φ \to ψ) \leftrightarrow (\neg ψ \to \forall x \neg φ)$