Theorem cbvriotadavw2

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

	Ref	Expression
Hypotheses	cbvriotadavw2.1	⊢ ( ( 𝜑 ∧ 𝑥 = 𝑦 ) → ( 𝜓 ↔ 𝜒 ) )
	cbvriotadavw2.2	⊢ ( ( 𝜑 ∧ 𝑥 = 𝑦 ) → 𝐴 = 𝐵 )
Assertion	cbvriotadavw2	⊢ ( 𝜑 → ( ℩ 𝑥 ∈ 𝐴 𝜓 ) = ( ℩ 𝑦 ∈ 𝐵 𝜒 ) )

Proof

Step	Hyp	Ref	Expression
1		cbvriotadavw2.1	⊢ ( ( 𝜑 ∧ 𝑥 = 𝑦 ) → ( 𝜓 ↔ 𝜒 ) )
2		cbvriotadavw2.2	⊢ ( ( 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	cbviotadavw	⊢ ( 𝜑 → ( ℩ 𝑥 ( 𝑥 ∈ 𝐴 ∧ 𝜓 ) ) = ( ℩ 𝑦 ( 𝑦 ∈ 𝐵 ∧ 𝜒 ) ) )
9		df-riota	⊢ ( ℩ 𝑥 ∈ 𝐴 𝜓 ) = ( ℩ 𝑥 ( 𝑥 ∈ 𝐴 ∧ 𝜓 ) )
10		df-riota	⊢ ( ℩ 𝑦 ∈ 𝐵 𝜒 ) = ( ℩ 𝑦 ( 𝑦 ∈ 𝐵 ∧ 𝜒 ) )
11	8 9 10	3eqtr4g	⊢ ( 𝜑 → ( ℩ 𝑥 ∈ 𝐴 𝜓 ) = ( ℩ 𝑦 ∈ 𝐵 𝜒 ) )