Theorem sb2vOLDOLD

Description: Obsolete version of sb2 as of 8-Jul-2023. Version of sb2 with a disjoint variable condition, which does not require ax-13 . (Contributed by BJ, 31-May-2019) (New usage is discouraged.) (Proof modification is discouraged.)

	Ref	Expression
Assertion	sb2vOLDOLD	$$⊢\forall x\left(x=y\to \phi \right)\to \left[y/x\right]\phi$$

Proof

Step	Hyp	Ref	Expression
1		sp	$$⊢\forall x\left(x=y\to \phi \right)\to \left(x=y\to \phi \right)$$
2		equs4v	$$⊢\forall x\left(x=y\to \phi \right)\to \exists x\left(x=y\wedge \phi \right)$$
3		dfsb1	$$⊢\left[y/x\right]\phi \leftrightarrow \left(\left(x=y\to \phi \right)\wedge \exists x\left(x=y\wedge \phi \right)\right)$$
4	1 2 3	sylanbrc	$$⊢\forall x\left(x=y\to \phi \right)\to \left[y/x\right]\phi$$