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)

	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		sb6	$$⊢\left[y/x\right]\phi \leftrightarrow \forall x\left(x=y\to \phi \right)$$
3	1	equsalvw	$$⊢\forall x\left(x=y\to \phi \right)\leftrightarrow \psi$$
4	2 3	bitri	$$⊢\left[y/x\right]\phi \leftrightarrow \psi$$