Theorem riotaeqbii 36654

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

Hypotheses

Ref	Expression
riotaeqbii.1	⊢ 𝐴 = 𝐵
riotaeqbii.2	⊢ (𝜑 ↔ 𝜓)

Assertion

Ref	Expression
riotaeqbii	⊢ (℩𝑥 ∈ 𝐴 𝜑) = (℩𝑥 ∈ 𝐵 𝜓)

Proof of Theorem riotaeqbii

Step	Hyp	Ref	Expression
1		riotaeqbii.1	. . . . 5 ⊢ 𝐴 = 𝐵
2	1	eleq2i 2853	. . . 4 ⊢ (𝑥 ∈ 𝐴 ↔ 𝑥 ∈ 𝐵)
3		riotaeqbii.2	. . . 4 ⊢ (𝜑 ↔ 𝜓)
4	2, 3	anbi12i 639	. . 3 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝜑) ↔ (𝑥 ∈ 𝐵 ∧ 𝜓))
5	4	iotabii 6521	. 2 ⊢ (℩𝑥(𝑥 ∈ 𝐴 ∧ 𝜑)) = (℩𝑥(𝑥 ∈ 𝐵 ∧ 𝜓))
6		df-riota 7367	. 2 ⊢ (℩𝑥 ∈ 𝐴 𝜑) = (℩𝑥(𝑥 ∈ 𝐴 ∧ 𝜑))
7		df-riota 7367	. 2 ⊢ (℩𝑥 ∈ 𝐵 𝜓) = (℩𝑥(𝑥 ∈ 𝐵 ∧ 𝜓))
8	5, 6, 7	3eqtr4i 2794	1 ⊢ (℩𝑥 ∈ 𝐴 𝜑) = (℩𝑥 ∈ 𝐵 𝜓)

Syntax hints:   ↔ wb 209   ∧ wa 400   = wceq 1568   ∈ wcel 2141  ℩cio 6490  ℩crio 7366

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1823  ax-4 1837  ax-5 1938  ax-6 1995  ax-7 2036  ax-8 2143  ax-9 2151  ax-ext 2733

This theorem depends on definitions:  df-bi 210  df-an 401  df-tru 1571  df-ex 1808  df-sb 2095  df-clab 2740  df-cleq 2753  df-clel 2836  df-v 3455  df-ss 3921  df-uni 4872  df-iota 6492  df-riota 7367

This theorem is referenced by:  riotaeqi  36655