Theorem cbvabdavw

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

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

Proof

Step	Hyp	Ref	Expression
1		cbvabdavw.1	⊢ ( ( 𝜑 ∧ 𝑥 = 𝑦 ) → ( 𝜓 ↔ 𝜒 ) )
2	1	cbvsbdavw	⊢ ( 𝜑 → ( [ 𝑡 / 𝑥 ] 𝜓 ↔ [ 𝑡 / 𝑦 ] 𝜒 ) )
3		df-clab	⊢ ( 𝑡 ∈ { 𝑥 ∣ 𝜓 } ↔ [ 𝑡 / 𝑥 ] 𝜓 )
4		df-clab	⊢ ( 𝑡 ∈ { 𝑦 ∣ 𝜒 } ↔ [ 𝑡 / 𝑦 ] 𝜒 )
5	2 3 4	3bitr4g	⊢ ( 𝜑 → ( 𝑡 ∈ { 𝑥 ∣ 𝜓 } ↔ 𝑡 ∈ { 𝑦 ∣ 𝜒 } ) )
6	5	eqrdv	⊢ ( 𝜑 → { 𝑥 ∣ 𝜓 } = { 𝑦 ∣ 𝜒 } )