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Metamath Proof Explorer

Theorem nfan1

Description: A closed form of nfan . (Contributed by Mario Carneiro, 3-Oct-2016) df-nf changed. (Revised by Wolf Lammen, 18-Sep-2021) (Proof shortened by Wolf Lammen, 7-Jul-2022)

	Ref	Expression
Hypotheses	nfim1.1	⊢ Ⅎ 𝑥 𝜑
	nfim1.2	⊢ ( 𝜑 → Ⅎ 𝑥 𝜓 )
Assertion	nfan1	⊢ Ⅎ 𝑥 ( 𝜑 ∧ 𝜓 )

Proof

Step	Hyp	Ref	Expression
1		nfim1.1	⊢ Ⅎ 𝑥 𝜑
2		nfim1.2	⊢ ( 𝜑 → Ⅎ 𝑥 𝜓 )
3		df-an	⊢ ( ( 𝜑 ∧ 𝜓 ) ↔ ¬ ( 𝜑 → ¬ 𝜓 ) )
4	2	nfnd	⊢ ( 𝜑 → Ⅎ 𝑥 ¬ 𝜓 )
5	1 4	nfim1	⊢ Ⅎ 𝑥 ( 𝜑 → ¬ 𝜓 )
6	5	nfn	⊢ Ⅎ 𝑥 ¬ ( 𝜑 → ¬ 𝜓 )
7	3 6	nfxfr	⊢ Ⅎ 𝑥 ( 𝜑 ∧ 𝜓 )