Theorem riotaeqbii 36242

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 2823	. . . 4 ⊢ (𝑥 ∈ 𝐴 ↔ 𝑥 ∈ 𝐵)
3		riotaeqbii.2	. . . 4 ⊢ (𝜑 ↔ 𝜓)
4	2, 3	anbi12i 628	. . 3 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝜑) ↔ (𝑥 ∈ 𝐵 ∧ 𝜓))
5	4	iotabii 6466	. 2 ⊢ (℩𝑥(𝑥 ∈ 𝐴 ∧ 𝜑)) = (℩𝑥(𝑥 ∈ 𝐵 ∧ 𝜓))
6		df-riota 7303	. 2 ⊢ (℩𝑥 ∈ 𝐴 𝜑) = (℩𝑥(𝑥 ∈ 𝐴 ∧ 𝜑))
7		df-riota 7303	. 2 ⊢ (℩𝑥 ∈ 𝐵 𝜓) = (℩𝑥(𝑥 ∈ 𝐵 ∧ 𝜓))
8	5, 6, 7	3eqtr4i 2764	1 ⊢ (℩𝑥 ∈ 𝐴 𝜑) = (℩𝑥 ∈ 𝐵 𝜓)

Syntax hints:   ↔ wb 206   ∧ wa 395   = wceq 1541   ∈ wcel 2111  ℩cio 6435  ℩crio 7302

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-ext 2703

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1544  df-ex 1781  df-sb 2068  df-clab 2710  df-cleq 2723  df-clel 2806  df-v 3438  df-ss 3914  df-uni 4857  df-iota 6437  df-riota 7303

This theorem is referenced by:  riotaeqi  36243