Theorem hban

Description: If x is not free in ph and ps , it is not free in ( ph /\ ps ) . (Contributed by NM, 14-May-1993) (Proof shortened by Wolf Lammen, 2-Jan-2018) Reduce axiom dependencies. (Revised by SN and Sky Winter, 22-Oct-2026)

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

Proof

Step	Hyp	Ref	Expression
1		hb.1	⊢ ( 𝜑 → ∀ 𝑥 𝜑 )
2		hb.2	⊢ ( 𝜓 → ∀ 𝑥 𝜓 )
3		pm3.2	⊢ ( 𝜑 → ( 𝜓 → ( 𝜑 ∧ 𝜓 ) ) )
4	1 2 3	alrimdh	⊢ ( 𝜑 → ( 𝜓 → ∀ 𝑥 ( 𝜑 ∧ 𝜓 ) ) )
5	4	imp	⊢ ( ( 𝜑 ∧ 𝜓 ) → ∀ 𝑥 ( 𝜑 ∧ 𝜓 ) )