Theorem bj-sbievwd

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

	Ref	Expression
Hypotheses	bj-sbievwd.nf0	⊢ ( 𝜑 → ∀ 𝑥 𝜑 )
	bj-sbievwd.nf	⊢ ( 𝜑 → Ⅎ' 𝑥 𝜒 )
	bj-sbievwd.is	⊢ ( ( 𝜑 ∧ 𝑥 = 𝑦 ) → ( 𝜓 ↔ 𝜒 ) )
Assertion	bj-sbievwd	⊢ ( 𝜑 → ( [ 𝑦 / 𝑥 ] 𝜓 ↔ 𝜒 ) )

Proof

Step	Hyp	Ref	Expression
1		bj-sbievwd.nf0	⊢ ( 𝜑 → ∀ 𝑥 𝜑 )
2		bj-sbievwd.nf	⊢ ( 𝜑 → Ⅎ' 𝑥 𝜒 )
3		bj-sbievwd.is	⊢ ( ( 𝜑 ∧ 𝑥 = 𝑦 ) → ( 𝜓 ↔ 𝜒 ) )
4		sb6	⊢ ( [ 𝑦 / 𝑥 ] 𝜓 ↔ ∀ 𝑥 ( 𝑥 = 𝑦 → 𝜓 ) )
5	1 2 3	bj-equsalvwd	⊢ ( 𝜑 → ( ∀ 𝑥 ( 𝑥 = 𝑦 → 𝜓 ) ↔ 𝜒 ) )
6	4 5	syl5bb	⊢ ( 𝜑 → ( [ 𝑦 / 𝑥 ] 𝜓 ↔ 𝜒 ) )