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Metamath Proof Explorer

Theorem hb3an

Description: If x is not free in ph , ps , and ch , it is not free in ( ph /\ ps /\ ch ) . (Contributed by NM, 14-Sep-2003) (Proof shortened by Wolf Lammen, 2-Jan-2018)

	Ref	Expression
Hypotheses	hb.1	⊢ ( 𝜑 → ∀ 𝑥 𝜑 )
	hb.2	⊢ ( 𝜓 → ∀ 𝑥 𝜓 )
	hb.3	⊢ ( 𝜒 → ∀ 𝑥 𝜒 )
Assertion	hb3an	⊢ ( ( 𝜑 ∧ 𝜓 ∧ 𝜒 ) → ∀ 𝑥 ( 𝜑 ∧ 𝜓 ∧ 𝜒 ) )

Proof

Step	Hyp	Ref	Expression
1		hb.1	⊢ ( 𝜑 → ∀ 𝑥 𝜑 )
2		hb.2	⊢ ( 𝜓 → ∀ 𝑥 𝜓 )
3		hb.3	⊢ ( 𝜒 → ∀ 𝑥 𝜒 )
4	1	nf5i	⊢ Ⅎ 𝑥 𝜑
5	2	nf5i	⊢ Ⅎ 𝑥 𝜓
6	3	nf5i	⊢ Ⅎ 𝑥 𝜒
7	4 5 6	nf3an	⊢ Ⅎ 𝑥 ( 𝜑 ∧ 𝜓 ∧ 𝜒 )
8	7	nf5ri	⊢ ( ( 𝜑 ∧ 𝜓 ∧ 𝜒 ) → ∀ 𝑥 ( 𝜑 ∧ 𝜓 ∧ 𝜒 ) )