This is an inofficial mirror of http://metamath.tirix.org for personal testing of a visualizer extension only.

Metamath Proof Explorer

Theorem hbex

Description: If x is not free in ph , then it is not free in E. y ph . (Contributed by NM, 12-Mar-1993) Reduce symbol count in nfex , hbex . (Revised by Wolf Lammen, 16-Oct-2021)

		Ref	Expression
	Hypothesis	hbex.1	$⊢ φ \to \forall x φ$
	Assertion	hbex	$⊢ \exists y φ \to \forall x \exists y φ$

Proof

Step	Hyp	Ref	Expression
1		hbex.1	$⊢ φ \to \forall x φ$
2	1	nf5i	$⊢ Ⅎ x φ$
3	2	nfex	$⊢ Ⅎ x \exists y φ$
4	3	nf5ri	$⊢ \exists y φ \to \forall x \exists y φ$