Theorem hbsbw

Description: If z is not free in ph , it is not free in [ y / x ] ph when y and z are distinct. Version of hbsb with a disjoint variable condition, which requires fewer axioms. (Contributed by NM, 12-Aug-1993) (Revised by Gino Giotto, 23-May-2024) (Proof shortened by Wolf Lammen, 14-May-2025)

	Ref	Expression
Hypothesis	hbsbw.1	|- ( ph -> A. z ph )
Assertion	hbsbw	|- ( [ y / x ] ph -> A. z [ y / x ] ph )

Proof

Step	Hyp	Ref	Expression
1		hbsbw.1	|- ( ph -> A. z ph )
2	1	sbimi	|- ( [ y / x ] ph -> [ y / x ] A. z ph )
3		sbal	|- ( [ y / x ] A. z ph <-> A. z [ y / x ] ph )
4	2 3	sylib	|- ( [ y / x ] ph -> A. z [ y / x ] ph )