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Metamath Proof Explorer

Theorem nfa1

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

		Ref	Expression
	Assertion	nfa1	⊢ Ⅎ 𝑥 ∀ 𝑥 𝜑

Proof

Step	Hyp	Ref	Expression
1		alex	⊢ ( ∀ 𝑥 𝜑 ↔ ¬ ∃ 𝑥 ¬ 𝜑 )
2		nfe1	⊢ Ⅎ 𝑥 ∃ 𝑥 ¬ 𝜑
3	2	nfn	⊢ Ⅎ 𝑥 ¬ ∃ 𝑥 ¬ 𝜑
4	1 3	nfxfr	⊢ Ⅎ 𝑥 ∀ 𝑥 𝜑