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Metamath Proof Explorer

Theorem nfan

Description: If x is not free in ph and ps , then it is not free in ( ph /\ ps ) . (Contributed by Mario Carneiro, 11-Aug-2016) (Proof shortened by Wolf Lammen, 13-Jan-2018) (Proof shortened by Wolf Lammen, 9-Oct-2021)

	Ref	Expression
Hypotheses	nfan.1	$⊢ Ⅎ x φ$
	nfan.2	$⊢ Ⅎ x ψ$
Assertion	nfan	$⊢ Ⅎ x (φ \land ψ)$

Proof

Step	Hyp	Ref	Expression
1		nfan.1	$⊢ Ⅎ x φ$
2		nfan.2	$⊢ Ⅎ x ψ$
3	1	a1i	$⊢ ⊤ \to Ⅎ x φ$
4	2	a1i	$⊢ ⊤ \to Ⅎ x ψ$
5	3 4	nfand	$⊢ ⊤ \to Ⅎ x (φ \land ψ)$
6	5	mptru	$⊢ Ⅎ x (φ \land ψ)$