Theorem cbviotadavw 36485

Description: Change bound variable in a description binder. Deduction form. (Contributed by GG, 14-Aug-2025.)

Hypothesis

Ref	Expression
cbviotadavw.1	⊢ ((𝜑 ∧ 𝑥 = 𝑦) → (𝜓 ↔ 𝜒))

Assertion

Ref	Expression
cbviotadavw	⊢ (𝜑 → (℩𝑥𝜓) = (℩𝑦𝜒))

Distinct variable groups:   𝜑,𝑥,𝑦   𝜓,𝑦   𝜒,𝑥

Allowed substitution hints:   𝜓(𝑥)   𝜒(𝑦)

Proof of Theorem cbviotadavw

Dummy variable 𝑡 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		cbviotadavw.1	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 = 𝑦) → (𝜓 ↔ 𝜒))
2	1	cbvabdavw 36472	. . . . 5 ⊢ (𝜑 → {𝑥 ∣ 𝜓} = {𝑦 ∣ 𝜒})
3	2	eqeq1d 2739	. . . 4 ⊢ (𝜑 → ({𝑥 ∣ 𝜓} = {𝑡} ↔ {𝑦 ∣ 𝜒} = {𝑡}))
4	3	abbidv 2803	. . 3 ⊢ (𝜑 → {𝑡 ∣ {𝑥 ∣ 𝜓} = {𝑡}} = {𝑡 ∣ {𝑦 ∣ 𝜒} = {𝑡}})
5	4	unieqd 4878	. 2 ⊢ (𝜑 → ∪ {𝑡 ∣ {𝑥 ∣ 𝜓} = {𝑡}} = ∪ {𝑡 ∣ {𝑦 ∣ 𝜒} = {𝑡}})
6		df-iota 6456	. 2 ⊢ (℩𝑥𝜓) = ∪ {𝑡 ∣ {𝑥 ∣ 𝜓} = {𝑡}}
7		df-iota 6456	. 2 ⊢ (℩𝑦𝜒) = ∪ {𝑡 ∣ {𝑦 ∣ 𝜒} = {𝑡}}
8	5, 6, 7	3eqtr4g 2797	1 ⊢ (𝜑 → (℩𝑥𝜓) = (℩𝑦𝜒))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1542  {cab 2715  {csn 4582  ∪ cuni 4865  ℩cio 6454

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-ext 2709

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1545  df-ex 1782  df-sb 2069  df-clab 2716  df-cleq 2729  df-clel 2812  df-v 3444  df-ss 3920  df-uni 4866  df-iota 6456

This theorem is referenced by:  cbvriotadavw  36486  cbvriotadavw2  36506