Theorem bj-hbs1

Description: Version of hbsb2 with a disjoint variable condition, which does not require ax-13 , and removal of ax-13 from hbs1 . (Contributed by BJ, 23-Jun-2019) (Proof modification is discouraged.)

	Ref	Expression
Assertion	bj-hbs1	$$⊢\left[y/x\right]\phi \to \forall x\left[y/x\right]\phi$$

Proof

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