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Metamath Proof Explorer

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 )