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	|- ( ph -> A. x ph )
	hb.2	|- ( ps -> A. x ps )
	hb.3	|- ( ch -> A. x ch )
Assertion	hb3an	|- ( ( ph /\ ps /\ ch ) -> A. x ( ph /\ ps /\ ch ) )

Proof

Step	Hyp	Ref	Expression
1		hb.1	|- ( ph -> A. x ph )
2		hb.2	|- ( ps -> A. x ps )
3		hb.3	|- ( ch -> A. x ch )
4	1	nf5i	|- F/ x ph
5	2	nf5i	|- F/ x ps
6	3	nf5i	|- F/ x ch
7	4 5 6	nf3an	|- F/ x ( ph /\ ps /\ ch )
8	7	nf5ri	|- ( ( ph /\ ps /\ ch ) -> A. x ( ph /\ ps /\ ch ) )