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Metamath Proof Explorer

Theorem nfnf1

Description: The setvar x is not free in F/ x ph . (Contributed by Mario Carneiro, 11-Aug-2016) Remove dependency on ax-12 . (Revised by Wolf Lammen, 12-Oct-2021)

		Ref	Expression
	Assertion	nfnf1	$⊢ Ⅎ x Ⅎ x φ$

Proof

Step	Hyp	Ref	Expression
1		df-nf	$⊢ Ⅎ x φ \leftrightarrow (\exists x φ \to \forall x φ)$
2		nfe1	$⊢ Ⅎ x \exists x φ$
3		nfa1	$⊢ Ⅎ x \forall x φ$
4	2 3	nfim	$⊢ Ⅎ x (\exists x φ \to \forall x φ)$
5	1 4	nfxfr	$⊢ Ⅎ x Ⅎ x φ$