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) Reduce axiom dependencies. (Revised by SN and Sky Winter, 22-Oct-2026)

	Ref	Expression
Hypotheses	hban.1	|- ( ph -> A. x ph )
	hban.2	|- ( ps -> A. x ps )
Assertion	hban	|- ( ( ph /\ ps ) -> A. x ( ph /\ ps ) )

Proof

Step	Hyp	Ref	Expression
1		hban.1	|- ( ph -> A. x ph )
2		hban.2	|- ( ps -> A. x ps )
3		pm3.2	|- ( ph -> ( ps -> ( ph /\ ps ) ) )
4	1 2 3	alrimdh	|- ( ph -> ( ps -> A. x ( ph /\ ps ) ) )
5	4	imp	|- ( ( ph /\ ps ) -> A. x ( ph /\ ps ) )