Theorem sbiev

Description: Conversion of implicit substitution to explicit substitution. Version of sbie with a disjoint variable condition, not requiring ax-13 . See sbievw for a version with a disjoint variable condition requiring fewer axioms. (Contributed by NM, 30-Jun-1994) (Revised by Wolf Lammen, 18-Jan-2023) Remove dependence on ax-10 and shorten proof. (Revised by BJ, 18-Jul-2023)

	Ref	Expression
Hypotheses	sbiev.1	$$⊢Ⅎx\psi$$
	sbiev.2	$$⊢x=y\to \left(\phi \leftrightarrow \psi \right)$$
Assertion	sbiev	$$⊢\left[y/x\right]\phi \leftrightarrow \psi$$

Proof

Step	Hyp	Ref	Expression
1		sbiev.1	$$⊢Ⅎx\psi$$
2		sbiev.2	$$⊢x=y\to \left(\phi \leftrightarrow \psi \right)$$
3		sb6	$$⊢\left[y/x\right]\phi \leftrightarrow \forall x\left(x=y\to \phi \right)$$
4	1 2	equsalv	$$⊢\forall x\left(x=y\to \phi \right)\leftrightarrow \psi$$
5	3 4	bitri	$$⊢\left[y/x\right]\phi \leftrightarrow \psi$$