Theorem bj-sbievwd

Description: Variant of sbievw . (Contributed by BJ, 7-Oct-2024)

	Ref	Expression
Hypotheses	bj-sbievwd.nf0	$$⊢\phi \to \forall x\phi$$
	bj-sbievwd.nf	$$⊢\phi \to Ⅎ\text{'}x\chi$$
	bj-sbievwd.is	$$⊢\left(\phi \wedge x=y\right)\to \left(\psi \leftrightarrow \chi \right)$$
Assertion	bj-sbievwd	$$⊢\phi \to \left(\left[y/x\right]\psi \leftrightarrow \chi \right)$$

Proof

Step	Hyp	Ref	Expression
1		bj-sbievwd.nf0	$$⊢\phi \to \forall x\phi$$
2		bj-sbievwd.nf	$$⊢\phi \to Ⅎ\text{'}x\chi$$
3		bj-sbievwd.is	$$⊢\left(\phi \wedge x=y\right)\to \left(\psi \leftrightarrow \chi \right)$$
4		sb6	$$⊢\left[y/x\right]\psi \leftrightarrow \forall x\left(x=y\to \psi \right)$$
5	1 2 3	bj-equsalvwd	$$⊢\phi \to \left(\forall x\left(x=y\to \psi \right)\leftrightarrow \chi \right)$$
6	4 5	syl5bb	$$⊢\phi \to \left(\left[y/x\right]\psi \leftrightarrow \chi \right)$$