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Metamath Proof Explorer

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	$⊢ \exists x φ \to φ$
	Assertion	nfexhe	$⊢ Ⅎ x \exists y φ$

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