Theorem cbvrabdavw2

Description: Change bound variable and domain in restricted class abstractions. Deduction form. (Contributed by GG, 14-Aug-2025)

	Ref	Expression
Hypotheses	cbvrabdavw2.1	⊢ ( ( 𝜑 ∧ 𝑥 = 𝑦 ) → ( 𝜓 ↔ 𝜒 ) )
	cbvrabdavw2.2	⊢ ( ( 𝜑 ∧ 𝑥 = 𝑦 ) → 𝐴 = 𝐵 )
Assertion	cbvrabdavw2	⊢ ( 𝜑 → { 𝑥 ∈ 𝐴 ∣ 𝜓 } = { 𝑦 ∈ 𝐵 ∣ 𝜒 } )

Proof

Step	Hyp	Ref	Expression
1		cbvrabdavw2.1	⊢ ( ( 𝜑 ∧ 𝑥 = 𝑦 ) → ( 𝜓 ↔ 𝜒 ) )
2		cbvrabdavw2.2	⊢ ( ( 𝜑 ∧ 𝑥 = 𝑦 ) → 𝐴 = 𝐵 )
3		eleq1w	⊢ ( 𝑥 = 𝑦 → ( 𝑥 ∈ 𝐴 ↔ 𝑦 ∈ 𝐴 ) )
4	3	adantl	⊢ ( ( 𝜑 ∧ 𝑥 = 𝑦 ) → ( 𝑥 ∈ 𝐴 ↔ 𝑦 ∈ 𝐴 ) )
5	2	eleq2d	⊢ ( ( 𝜑 ∧ 𝑥 = 𝑦 ) → ( 𝑦 ∈ 𝐴 ↔ 𝑦 ∈ 𝐵 ) )
6	4 5	bitrd	⊢ ( ( 𝜑 ∧ 𝑥 = 𝑦 ) → ( 𝑥 ∈ 𝐴 ↔ 𝑦 ∈ 𝐵 ) )
7	6 1	anbi12d	⊢ ( ( 𝜑 ∧ 𝑥 = 𝑦 ) → ( ( 𝑥 ∈ 𝐴 ∧ 𝜓 ) ↔ ( 𝑦 ∈ 𝐵 ∧ 𝜒 ) ) )
8	7	cbvabdavw	⊢ ( 𝜑 → { 𝑥 ∣ ( 𝑥 ∈ 𝐴 ∧ 𝜓 ) } = { 𝑦 ∣ ( 𝑦 ∈ 𝐵 ∧ 𝜒 ) } )
9		df-rab	⊢ { 𝑥 ∈ 𝐴 ∣ 𝜓 } = { 𝑥 ∣ ( 𝑥 ∈ 𝐴 ∧ 𝜓 ) }
10		df-rab	⊢ { 𝑦 ∈ 𝐵 ∣ 𝜒 } = { 𝑦 ∣ ( 𝑦 ∈ 𝐵 ∧ 𝜒 ) }
11	8 9 10	3eqtr4g	⊢ ( 𝜑 → { 𝑥 ∈ 𝐴 ∣ 𝜓 } = { 𝑦 ∈ 𝐵 ∣ 𝜒 } )