Theorem nf3an

Description: If x is not free in ph , ps , and ch , then it is not free in ( ph /\ ps /\ ch ) . (Contributed by Mario Carneiro, 11-Aug-2016)

	Ref	Expression
Hypotheses	nfan.1	⊢ Ⅎ 𝑥 𝜑
	nfan.2	⊢ Ⅎ 𝑥 𝜓
	nfan.3	⊢ Ⅎ 𝑥 𝜒
Assertion	nf3an	⊢ Ⅎ 𝑥 ( 𝜑 ∧ 𝜓 ∧ 𝜒 )

Proof

Step	Hyp	Ref	Expression
1		nfan.1	⊢ Ⅎ 𝑥 𝜑
2		nfan.2	⊢ Ⅎ 𝑥 𝜓
3		nfan.3	⊢ Ⅎ 𝑥 𝜒
4		df-3an	⊢ ( ( 𝜑 ∧ 𝜓 ∧ 𝜒 ) ↔ ( ( 𝜑 ∧ 𝜓 ) ∧ 𝜒 ) )
5	1 2	nfan	⊢ Ⅎ 𝑥 ( 𝜑 ∧ 𝜓 )
6	5 3	nfan	⊢ Ⅎ 𝑥 ( ( 𝜑 ∧ 𝜓 ) ∧ 𝜒 )
7	4 6	nfxfr	⊢ Ⅎ 𝑥 ( 𝜑 ∧ 𝜓 ∧ 𝜒 )