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	|- ( ph -> A. x ph )
	hb.2	|- ( ps -> A. x ps )
Assertion	hban	|- ( ( ph /\ ps ) -> A. x ( ph /\ ps ) )

Proof

Step	Hyp	Ref	Expression
1		hb.1	|- ( ph -> A. x ph )
2		hb.2	|- ( ps -> A. x ps )
3	1	nf5i	|- F/ x ph
4	2	nf5i	|- F/ x ps
5	3 4	nfan	|- F/ x ( ph /\ ps )
6	5	nf5ri	|- ( ( ph /\ ps ) -> A. x ( ph /\ ps ) )