Theorem cbvrabdavw

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

	Ref	Expression
Hypothesis	cbvrabdavw.1	⊢ ( ( 𝜑 ∧ 𝑥 = 𝑦 ) → ( 𝜓 ↔ 𝜒 ) )
Assertion	cbvrabdavw	⊢ ( 𝜑 → { 𝑥 ∈ 𝐴 ∣ 𝜓 } = { 𝑦 ∈ 𝐴 ∣ 𝜒 } )

Proof

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