Theorem nfexhe

Description: Version of nfex with the existential dual to the 'h' hypothesis, avoiding ax-12 . (Contributed by SN, 11-Feb-2026)

	Ref	Expression
Hypothesis	nfexhe.1	$$⊢\exists x\phi \to \phi$$
Assertion	nfexhe	$$⊢Ⅎx\exists y\phi$$

Proof

Step	Hyp	Ref	Expression
1		nfexhe.1	$$⊢\exists x\phi \to \phi$$
2		hbe1	$$⊢\exists x\exists y\phi \to \forall x\exists x\exists y\phi$$
3		excomim	$$⊢\exists x\exists y\phi \to \exists y\exists x\phi$$
4	1	eximi	$$⊢\exists y\exists x\phi \to \exists y\phi$$
5	3 4	syl	$$⊢\exists x\exists y\phi \to \exists y\phi$$
6	2 5	alrimih	$$⊢\exists x\exists y\phi \to \forall x\exists y\phi$$
7	6	nfi	$$⊢Ⅎx\exists y\phi$$