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	$$⊢\phi \to \forall x\phi$$
	hb.2	$$⊢\psi \to \forall x\psi$$
	hb.3	$$⊢\chi \to \forall x\chi$$
Assertion	hb3an	$$⊢\left(\phi \wedge \psi \wedge \chi \right)\to \forall x\left(\phi \wedge \psi \wedge \chi \right)$$

Proof

Step	Hyp	Ref	Expression
1		hb.1	$$⊢\phi \to \forall x\phi$$
2		hb.2	$$⊢\psi \to \forall x\psi$$
3		hb.3	$$⊢\chi \to \forall x\chi$$
4	1	nf5i	$$⊢Ⅎx\phi$$
5	2	nf5i	$$⊢Ⅎx\psi$$
6	3	nf5i	$$⊢Ⅎx\chi$$
7	4 5 6	nf3an	$$⊢Ⅎx\left(\phi \wedge \psi \wedge \chi \right)$$
8	7	nf5ri	$$⊢\left(\phi \wedge \psi \wedge \chi \right)\to \forall x\left(\phi \wedge \psi \wedge \chi \right)$$