Theorem cbvsbcdavw2

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

	Ref	Expression
Hypotheses	cbvsbcdavw2.1	⊢ ( 𝜑 → 𝐴 = 𝐵 )
	cbvsbcdavw2.2	⊢ ( ( 𝜑 ∧ 𝑥 = 𝑦 ) → ( 𝜓 ↔ 𝜒 ) )
Assertion	cbvsbcdavw2	⊢ ( 𝜑 → ( [ 𝐴 / 𝑥 ] 𝜓 ↔ [ 𝐵 / 𝑦 ] 𝜒 ) )

Proof

Step	Hyp	Ref	Expression
1		cbvsbcdavw2.1	⊢ ( 𝜑 → 𝐴 = 𝐵 )
2		cbvsbcdavw2.2	⊢ ( ( 𝜑 ∧ 𝑥 = 𝑦 ) → ( 𝜓 ↔ 𝜒 ) )
3	2	cbvabdavw	⊢ ( 𝜑 → { 𝑥 ∣ 𝜓 } = { 𝑦 ∣ 𝜒 } )
4	1 3	eleq12d	⊢ ( 𝜑 → ( 𝐴 ∈ { 𝑥 ∣ 𝜓 } ↔ 𝐵 ∈ { 𝑦 ∣ 𝜒 } ) )
5		df-sbc	⊢ ( [ 𝐴 / 𝑥 ] 𝜓 ↔ 𝐴 ∈ { 𝑥 ∣ 𝜓 } )
6		df-sbc	⊢ ( [ 𝐵 / 𝑦 ] 𝜒 ↔ 𝐵 ∈ { 𝑦 ∣ 𝜒 } )
7	4 5 6	3bitr4g	⊢ ( 𝜑 → ( [ 𝐴 / 𝑥 ] 𝜓 ↔ [ 𝐵 / 𝑦 ] 𝜒 ) )