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) Reduce axiom dependencies. (Revised by SN and Sky Winter, 23-Oct-2026)

	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		df-3an	⊢ ( ( 𝜑 ∧ 𝜓 ∧ 𝜒 ) ↔ ( ( 𝜑 ∧ 𝜓 ) ∧ 𝜒 ) )
5	1 2	hban	⊢ ( ( 𝜑 ∧ 𝜓 ) → ∀ 𝑥 ( 𝜑 ∧ 𝜓 ) )
6	5 3	hban	⊢ ( ( ( 𝜑 ∧ 𝜓 ) ∧ 𝜒 ) → ∀ 𝑥 ( ( 𝜑 ∧ 𝜓 ) ∧ 𝜒 ) )
7	4 6	hbxfrbi	⊢ ( ( 𝜑 ∧ 𝜓 ∧ 𝜒 ) → ∀ 𝑥 ( 𝜑 ∧ 𝜓 ∧ 𝜒 ) )