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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	$⊢ φ \to Ⅎ x ψ$
Assertion	nfan1	$⊢ Ⅎ x (φ \land ψ)$

Proof

Step	Hyp	Ref	Expression
1		nfim1.1	$⊢ Ⅎ x φ$
2		nfim1.2	$⊢ φ \to Ⅎ x ψ$
3		df-an	$⊢ (φ \land ψ) \leftrightarrow \neg (φ \to \neg ψ)$
4	2	nfnd	$⊢ φ \to Ⅎ x \neg ψ$
5	1 4	nfim1	$⊢ Ⅎ x (φ \to \neg ψ)$
6	5	nfn	$⊢ Ⅎ x \neg (φ \to \neg ψ)$
7	3 6	nfxfr	$⊢ Ⅎ x (φ \land ψ)$