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) Reduce axiom dependencies. (Revised by SN and Sky Winter, 23-Oct-2026)

	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		df-3an	|- ( ( ph /\ ps /\ ch ) <-> ( ( ph /\ ps ) /\ ch ) )
5	1 2	hban	|- ( ( ph /\ ps ) -> A. x ( ph /\ ps ) )
6	5 3	hban	|- ( ( ( ph /\ ps ) /\ ch ) -> A. x ( ( ph /\ ps ) /\ ch ) )
7	4 6	hbxfrbi	|- ( ( ph /\ ps /\ ch ) -> A. x ( ph /\ ps /\ ch ) )