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	$$⊢\phi \to \forall z\phi$$
Assertion	hbsbw	$$⊢\left[y/x\right]\phi \to \forall z\left[y/x\right]\phi$$

Proof

Step	Hyp	Ref	Expression
1		hbsbw.1	$$⊢\phi \to \forall z\phi$$
2	1	sbimi	$$⊢\left[y/x\right]\phi \to \left[y/x\right]\forall z\phi$$
3		sbal	$$⊢\left[y/x\right]\forall z\phi \leftrightarrow \forall z\left[y/x\right]\phi$$
4	2 3	sylib	$$⊢\left[y/x\right]\phi \to \forall z\left[y/x\right]\phi$$