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	⊢ Ⅎ 𝑥 𝜓
	bj-sbievv.nfy	⊢ Ⅎ 𝑦 𝜑
	bj-sbievv.is	⊢ ( 𝑥 = 𝑦 → ( 𝜑 ↔ 𝜓 ) )
Assertion	bj-sbievv	⊢ ( [ 𝑦 / 𝑥 ] 𝜑 ↔ 𝜓 )

Proof

Step	Hyp	Ref	Expression
1		bj-sbievv.nfx	⊢ Ⅎ 𝑥 𝜓
2		bj-sbievv.nfy	⊢ Ⅎ 𝑦 𝜑
3		bj-sbievv.is	⊢ ( 𝑥 = 𝑦 → ( 𝜑 ↔ 𝜓 ) )
4	2	sb6f	⊢ ( [ 𝑦 / 𝑥 ] 𝜑 ↔ ∀ 𝑥 ( 𝑥 = 𝑦 → 𝜑 ) )
5	1 3	equsal	⊢ ( ∀ 𝑥 ( 𝑥 = 𝑦 → 𝜑 ) ↔ 𝜓 )
6	4 5	bitri	⊢ ( [ 𝑦 / 𝑥 ] 𝜑 ↔ 𝜓 )