Theorem cbvriotavw2

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

	Ref	Expression
Hypotheses	cbvriotavw2.1	⊢ ( 𝑥 = 𝑦 → 𝐴 = 𝐵 )
	cbvriotavw2.2	⊢ ( 𝑥 = 𝑦 → ( 𝜑 ↔ 𝜓 ) )
Assertion	cbvriotavw2	⊢ ( ℩ 𝑥 ∈ 𝐴 𝜑 ) = ( ℩ 𝑦 ∈ 𝐵 𝜓 )

Proof

Step	Hyp	Ref	Expression
1		cbvriotavw2.1	⊢ ( 𝑥 = 𝑦 → 𝐴 = 𝐵 )
2		cbvriotavw2.2	⊢ ( 𝑥 = 𝑦 → ( 𝜑 ↔ 𝜓 ) )
3		id	⊢ ( 𝑥 = 𝑦 → 𝑥 = 𝑦 )
4	3 1	eleq12d	⊢ ( 𝑥 = 𝑦 → ( 𝑥 ∈ 𝐴 ↔ 𝑦 ∈ 𝐵 ) )
5	4 2	anbi12d	⊢ ( 𝑥 = 𝑦 → ( ( 𝑥 ∈ 𝐴 ∧ 𝜑 ) ↔ ( 𝑦 ∈ 𝐵 ∧ 𝜓 ) ) )
6	5	cbviotavw	⊢ ( ℩ 𝑥 ( 𝑥 ∈ 𝐴 ∧ 𝜑 ) ) = ( ℩ 𝑦 ( 𝑦 ∈ 𝐵 ∧ 𝜓 ) )
7		df-riota	⊢ ( ℩ 𝑥 ∈ 𝐴 𝜑 ) = ( ℩ 𝑥 ( 𝑥 ∈ 𝐴 ∧ 𝜑 ) )
8		df-riota	⊢ ( ℩ 𝑦 ∈ 𝐵 𝜓 ) = ( ℩ 𝑦 ( 𝑦 ∈ 𝐵 ∧ 𝜓 ) )
9	6 7 8	3eqtr4i	⊢ ( ℩ 𝑥 ∈ 𝐴 𝜑 ) = ( ℩ 𝑦 ∈ 𝐵 𝜓 )