Theorem cbvcsbdavw

Description: Change bound variable of a proper substitution into a class. Deduction form. (Contributed by GG, 14-Aug-2025)

	Ref	Expression
Hypothesis	cbvcsbdavw.1	⊢ ( ( 𝜑 ∧ 𝑥 = 𝑦 ) → 𝐵 = 𝐶 )
Assertion	cbvcsbdavw	⊢ ( 𝜑 → ⦋ 𝐴 / 𝑥 ⦌ 𝐵 = ⦋ 𝐴 / 𝑦 ⦌ 𝐶 )

Proof

Step	Hyp	Ref	Expression
1		cbvcsbdavw.1	⊢ ( ( 𝜑 ∧ 𝑥 = 𝑦 ) → 𝐵 = 𝐶 )
2	1	eleq2d	⊢ ( ( 𝜑 ∧ 𝑥 = 𝑦 ) → ( 𝑡 ∈ 𝐵 ↔ 𝑡 ∈ 𝐶 ) )
3	2	cbvsbcdavw	⊢ ( 𝜑 → ( [ 𝐴 / 𝑥 ] 𝑡 ∈ 𝐵 ↔ [ 𝐴 / 𝑦 ] 𝑡 ∈ 𝐶 ) )
4	3	abbidv	⊢ ( 𝜑 → { 𝑡 ∣ [ 𝐴 / 𝑥 ] 𝑡 ∈ 𝐵 } = { 𝑡 ∣ [ 𝐴 / 𝑦 ] 𝑡 ∈ 𝐶 } )
5		df-csb	⊢ ⦋ 𝐴 / 𝑥 ⦌ 𝐵 = { 𝑡 ∣ [ 𝐴 / 𝑥 ] 𝑡 ∈ 𝐵 }
6		df-csb	⊢ ⦋ 𝐴 / 𝑦 ⦌ 𝐶 = { 𝑡 ∣ [ 𝐴 / 𝑦 ] 𝑡 ∈ 𝐶 }
7	4 5 6	3eqtr4g	⊢ ( 𝜑 → ⦋ 𝐴 / 𝑥 ⦌ 𝐵 = ⦋ 𝐴 / 𝑦 ⦌ 𝐶 )