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Metamath Proof Explorer

Theorem alimex

Description: An equivalence between an implication with a universally quantified consequent and an implication with an existentially quantified antecedent. 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 eximal . (Contributed by BJ, 12-May-2019)

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

Proof

Step	Hyp	Ref	Expression
1		alex	$⊢ \forall x ψ \leftrightarrow \neg \exists x \neg ψ$
2	1	imbi2i	$⊢ (φ \to \forall x ψ) \leftrightarrow (φ \to \neg \exists x \neg ψ)$
3		con2b	$⊢ (φ \to \neg \exists x \neg ψ) \leftrightarrow (\exists x \neg ψ \to \neg φ)$
4	2 3	bitri	$⊢ (φ \to \forall x ψ) \leftrightarrow (\exists x \neg ψ \to \neg φ)$