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	$$⊢\phi \to \forall x\phi$$
	hb.2	$$⊢\psi \to \forall x\psi$$
Assertion	hban	$$⊢\left(\phi \wedge \psi \right)\to \forall x\left(\phi \wedge \psi \right)$$

Proof

Step	Hyp	Ref	Expression
1		hb.1	$$⊢\phi \to \forall x\phi$$
2		hb.2	$$⊢\psi \to \forall x\psi$$
3		pm3.2	$$⊢\phi \to \left(\psi \to \left(\phi \wedge \psi \right)\right)$$
4	1 2 3	alrimdh	$$⊢\phi \to \left(\psi \to \forall x\left(\phi \wedge \psi \right)\right)$$
5	4	imp	$$⊢\left(\phi \wedge \psi \right)\to \forall x\left(\phi \wedge \psi \right)$$