Theorem riotaeqbii

Description: Equivalent wff's and equal domains yield equal restricted iotas. Inference version. (Contributed by GG, 1-Sep-2025)

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

Proof

Step	Hyp	Ref	Expression
1		riotaeqbii.1	⊢ 𝐴 = 𝐵
2		riotaeqbii.2	⊢ ( 𝜑 ↔ 𝜓 )
3	1	eleq2i	⊢ ( 𝑥 ∈ 𝐴 ↔ 𝑥 ∈ 𝐵 )
4	3 2	anbi12i	⊢ ( ( 𝑥 ∈ 𝐴 ∧ 𝜑 ) ↔ ( 𝑥 ∈ 𝐵 ∧ 𝜓 ) )
5	4	iotabii	⊢ ( ℩ 𝑥 ( 𝑥 ∈ 𝐴 ∧ 𝜑 ) ) = ( ℩ 𝑥 ( 𝑥 ∈ 𝐵 ∧ 𝜓 ) )
6		df-riota	⊢ ( ℩ 𝑥 ∈ 𝐴 𝜑 ) = ( ℩ 𝑥 ( 𝑥 ∈ 𝐴 ∧ 𝜑 ) )
7		df-riota	⊢ ( ℩ 𝑥 ∈ 𝐵 𝜓 ) = ( ℩ 𝑥 ( 𝑥 ∈ 𝐵 ∧ 𝜓 ) )
8	5 6 7	3eqtr4i	⊢ ( ℩ 𝑥 ∈ 𝐴 𝜑 ) = ( ℩ 𝑥 ∈ 𝐵 𝜓 )