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)

	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	1	nf5i	$$⊢Ⅎx\phi$$
4	2	nf5i	$$⊢Ⅎx\psi$$
5	3 4	nfan	$$⊢Ⅎx\left(\phi \wedge \psi \right)$$
6	5	nf5ri	$$⊢\left(\phi \wedge \psi \right)\to \forall x\left(\phi \wedge \psi \right)$$