Theorem sbievw

Description: Conversion of implicit substitution to explicit substitution. Version of sbie and sbiev with more disjoint variable conditions, requiring fewer axioms. (Contributed by NM, 30-Jun-1994) (Revised by BJ, 18-Jul-2023) (Proof shortened by SN, 24-Aug-2025)

	Ref	Expression
Hypothesis	sbievw.is	$$⊢x=y\to \left(\phi \leftrightarrow \psi \right)$$
Assertion	sbievw	$$⊢\left[y/x\right]\phi \leftrightarrow \psi$$

Proof

Step	Hyp	Ref	Expression
1		sbievw.is	$$⊢x=y\to \left(\phi \leftrightarrow \psi \right)$$
2	1	sbbiiev	$$⊢\left[y/x\right]\phi \leftrightarrow \left[y/x\right]\psi$$
3		sbv	$$⊢\left[y/x\right]\psi \leftrightarrow \psi$$
4	2 3	bitri	$$⊢\left[y/x\right]\phi \leftrightarrow \psi$$