Theorem bj-sbievv

Description: Version of sbie with a second nonfreeness hypothesis and shorter proof. (Contributed by BJ, 18-Jul-2023) (Proof modification is discouraged.)

	Ref	Expression
Hypotheses	bj-sbievv.nfx	$$⊢Ⅎx\psi$$
	bj-sbievv.nfy	$$⊢Ⅎy\phi$$
	bj-sbievv.is	$$⊢x=y\to \left(\phi \leftrightarrow \psi \right)$$
Assertion	bj-sbievv	$$⊢\left[y/x\right]\phi \leftrightarrow \psi$$

Proof

Step	Hyp	Ref	Expression
1		bj-sbievv.nfx	$$⊢Ⅎx\psi$$
2		bj-sbievv.nfy	$$⊢Ⅎy\phi$$
3		bj-sbievv.is	$$⊢x=y\to \left(\phi \leftrightarrow \psi \right)$$
4	2	sb6f	$$⊢\left[y/x\right]\phi \leftrightarrow \forall x\left(x=y\to \phi \right)$$
5	1 3	equsal	$$⊢\forall x\left(x=y\to \phi \right)\leftrightarrow \psi$$
6	4 5	bitri	$$⊢\left[y/x\right]\phi \leftrightarrow \psi$$