Theorem bj-hbsb2av

Description: Version of hbsb2a with a disjoint variable condition, which does not require ax-13 . (Contributed by BJ, 11-Sep-2019) (Proof modification is discouraged.)

	Ref	Expression
Assertion	bj-hbsb2av	$$⊢\left[y/x\right]\forall y\phi \to \forall x\left[y/x\right]\phi$$

Proof

Step	Hyp	Ref	Expression
1		sb4av	$$⊢\left[y/x\right]\forall y\phi \to \forall x\left(x=y\to \phi \right)$$
2		sb6	$$⊢\left[y/x\right]\phi \leftrightarrow \forall x\left(x=y\to \phi \right)$$
3	2	biimpri	$$⊢\forall x\left(x=y\to \phi \right)\to \left[y/x\right]\phi$$
4	3	axc4i	$$⊢\forall x\left(x=y\to \phi \right)\to \forall x\left[y/x\right]\phi$$
5	1 4	syl	$$⊢\left[y/x\right]\forall y\phi \to \forall x\left[y/x\right]\phi$$