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 non-freeness. See also alimex . (Contributed by BJ, 12-May-2019)

	Ref	Expression
Assertion	eximal	$$⊢\left(\exists x\phi \to \psi \right)\leftrightarrow \left(\neg \psi \to \forall x\neg \phi \right)$$

Proof

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