Theorem cbviotadavw 36208

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 36195	. . . . 5 ⊢ (𝜑 → {𝑥 ∣ 𝜓} = {𝑦 ∣ 𝜒})
3	2	eqeq1d 2736	. . . 4 ⊢ (𝜑 → ({𝑥 ∣ 𝜓} = {𝑡} ↔ {𝑦 ∣ 𝜒} = {𝑡}))
4	3	abbidv 2800	. . 3 ⊢ (𝜑 → {𝑡 ∣ {𝑥 ∣ 𝜓} = {𝑡}} = {𝑡 ∣ {𝑦 ∣ 𝜒} = {𝑡}})
5	4	unieqd 4893	. 2 ⊢ (𝜑 → ∪ {𝑡 ∣ {𝑥 ∣ 𝜓} = {𝑡}} = ∪ {𝑡 ∣ {𝑦 ∣ 𝜒} = {𝑡}})
6		df-iota 6480	. 2 ⊢ (℩𝑥𝜓) = ∪ {𝑡 ∣ {𝑥 ∣ 𝜓} = {𝑡}}
7		df-iota 6480	. 2 ⊢ (℩𝑦𝜒) = ∪ {𝑡 ∣ {𝑦 ∣ 𝜒} = {𝑡}}
8	5, 6, 7	3eqtr4g 2794	1 ⊢ (𝜑 → (℩𝑥𝜓) = (℩𝑦𝜒))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1539  {cab 2712  {csn 4599  ∪ cuni 4880  ℩cio 6478

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-ext 2706

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1542  df-ex 1779  df-sb 2064  df-clab 2713  df-cleq 2726  df-clel 2808  df-v 3459  df-ss 3941  df-uni 4881  df-iota 6480

This theorem is referenced by:  cbvriotadavw  36209  cbvriotadavw2  36229