Theorem cbvriotavw2 36464

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

Hypotheses

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

Assertion

Ref	Expression
cbvriotavw2	⊢ (℩𝑥 ∈ 𝐴 𝜑) = (℩𝑦 ∈ 𝐵 𝜓)

Distinct variable groups:   𝑥,𝑦   𝜑,𝑦   𝜓,𝑥   𝑦,𝐴   𝑥,𝐵

Allowed substitution hints:   𝜑(𝑥)   𝜓(𝑦)   𝐴(𝑥)   𝐵(𝑦)

Proof of Theorem cbvriotavw2

Step	Hyp	Ref	Expression
1		id 22	. . . . 5 ⊢ (𝑥 = 𝑦 → 𝑥 = 𝑦)
2		cbvriotavw2.1	. . . . 5 ⊢ (𝑥 = 𝑦 → 𝐴 = 𝐵)
3	1, 2	eleq12d 2833	. . . 4 ⊢ (𝑥 = 𝑦 → (𝑥 ∈ 𝐴 ↔ 𝑦 ∈ 𝐵))
4		cbvriotavw2.2	. . . 4 ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓))
5	3, 4	anbi12d 638	. . 3 ⊢ (𝑥 = 𝑦 → ((𝑥 ∈ 𝐴 ∧ 𝜑) ↔ (𝑦 ∈ 𝐵 ∧ 𝜓)))
6	5	cbviotavw 6449	. 2 ⊢ (℩𝑥(𝑥 ∈ 𝐴 ∧ 𝜑)) = (℩𝑦(𝑦 ∈ 𝐵 ∧ 𝜓))
7		df-riota 7313	. 2 ⊢ (℩𝑥 ∈ 𝐴 𝜑) = (℩𝑥(𝑥 ∈ 𝐴 ∧ 𝜑))
8		df-riota 7313	. 2 ⊢ (℩𝑦 ∈ 𝐵 𝜓) = (℩𝑦(𝑦 ∈ 𝐵 ∧ 𝜓))
9	6, 7, 8	3eqtr4i 2772	1 ⊢ (℩𝑥 ∈ 𝐴 𝜑) = (℩𝑦 ∈ 𝐵 𝜓)

Syntax hints:   → wi 4   ↔ wb 207   ∧ wa 396   = wceq 1547   ∈ wcel 2119  ℩cio 6439  ℩crio 7312

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-ext 2711

This theorem depends on definitions:  df-bi 208  df-an 397  df-tru 1550  df-ex 1787  df-sb 2074  df-clab 2718  df-cleq 2731  df-clel 2814  df-v 3433  df-ss 3900  df-uni 4839  df-iota 6441  df-riota 7313

This theorem is referenced by: (None)