Theorem bj-exexalal

Description: A lemma for changing bound variables. Only the forward implication is intuitionistic. (Contributed by BJ, 14-Mar-2026)

	Ref	Expression
Assertion	bj-exexalal	$$⊢\left(\exists x\phi \to \exists y\psi \right)\leftrightarrow \left(\forall y\neg \psi \to \forall x\neg \phi \right)$$

Proof

Step	Hyp	Ref	Expression
1		con34b	$$⊢\left(\exists x\phi \to \exists y\psi \right)\leftrightarrow \left(\neg \exists y\psi \to \neg \exists x\phi \right)$$
2		alnex	$$⊢\forall y\neg \psi \leftrightarrow \neg \exists y\psi$$
3		alnex	$$⊢\forall x\neg \phi \leftrightarrow \neg \exists x\phi$$
4	2 3	imbi12i	$$⊢\left(\forall y\neg \psi \to \forall x\neg \phi \right)\leftrightarrow \left(\neg \exists y\psi \to \neg \exists x\phi \right)$$
5	1 4	bitr4i	$$⊢\left(\exists x\phi \to \exists y\psi \right)\leftrightarrow \left(\forall y\neg \psi \to \forall x\neg \phi \right)$$