Theorem sbiedwOLD

Description: Obsolete version of sbiedw as of 28-Jan-2024. (Contributed by Gino Giotto, 10-Jan-2024) (Proof modification is discouraged.) (New usage is discouraged.)

	Ref	Expression
Hypotheses	sbiedw.1	⊢ Ⅎ 𝑥 𝜑
	sbiedw.2	⊢ ( 𝜑 → Ⅎ 𝑥 𝜒 )
	sbiedw.3	⊢ ( 𝜑 → ( 𝑥 = 𝑦 → ( 𝜓 ↔ 𝜒 ) ) )
Assertion	sbiedwOLD	⊢ ( 𝜑 → ( [ 𝑦 / 𝑥 ] 𝜓 ↔ 𝜒 ) )

Proof

Step	Hyp	Ref	Expression
1		sbiedw.1	⊢ Ⅎ 𝑥 𝜑
2		sbiedw.2	⊢ ( 𝜑 → Ⅎ 𝑥 𝜒 )
3		sbiedw.3	⊢ ( 𝜑 → ( 𝑥 = 𝑦 → ( 𝜓 ↔ 𝜒 ) ) )
4	1	sbrim	⊢ ( [ 𝑦 / 𝑥 ] ( 𝜑 → 𝜓 ) ↔ ( 𝜑 → [ 𝑦 / 𝑥 ] 𝜓 ) )
5	1 2	nfim1	⊢ Ⅎ 𝑥 ( 𝜑 → 𝜒 )
6	3	com12	⊢ ( 𝑥 = 𝑦 → ( 𝜑 → ( 𝜓 ↔ 𝜒 ) ) )
7	6	pm5.74d	⊢ ( 𝑥 = 𝑦 → ( ( 𝜑 → 𝜓 ) ↔ ( 𝜑 → 𝜒 ) ) )
8	5 7	sbiev	⊢ ( [ 𝑦 / 𝑥 ] ( 𝜑 → 𝜓 ) ↔ ( 𝜑 → 𝜒 ) )
9	4 8	bitr3i	⊢ ( ( 𝜑 → [ 𝑦 / 𝑥 ] 𝜓 ) ↔ ( 𝜑 → 𝜒 ) )
10	9	pm5.74ri	⊢ ( 𝜑 → ( [ 𝑦 / 𝑥 ] 𝜓 ↔ 𝜒 ) )