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Metamath Proof Explorer

Theorem nfnf

Description: If x is not free in ph , then it is not free in F/ y ph . (Contributed by Mario Carneiro, 11-Aug-2016) (Proof shortened by Wolf Lammen, 30-Dec-2017)

		Ref	Expression
	Hypothesis	nfnf.1	$⊢ Ⅎ x φ$
	Assertion	nfnf	$⊢ Ⅎ x Ⅎ y φ$

Proof

Step	Hyp	Ref	Expression
1		nfnf.1	$⊢ Ⅎ x φ$
2		df-nf	$⊢ Ⅎ y φ \leftrightarrow (\exists y φ \to \forall y φ)$
3	1	nfex	$⊢ Ⅎ x \exists y φ$
4	1	nfal	$⊢ Ⅎ x \forall y φ$
5	3 4	nfim	$⊢ Ⅎ x (\exists y φ \to \forall y φ)$
6	2 5	nfxfr	$⊢ Ⅎ x Ⅎ y φ$