Theorem cbviotadavw 37273

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 37260	. . . . 5 ⊢ (𝜑 → {𝑥 ∣ 𝜓} = {𝑦 ∣ 𝜒})
3	2	eqeq1d 2765	. . . 4 ⊢ (𝜑 → ({𝑥 ∣ 𝜓} = {𝑡} ↔ {𝑦 ∣ 𝜒} = {𝑡}))
4	3	abbidv 2829	. . 3 ⊢ (𝜑 → {𝑡 ∣ {𝑥 ∣ 𝜓} = {𝑡}} = {𝑡 ∣ {𝑦 ∣ 𝜒} = {𝑡}})
5	4	unieqd 4885	. 2 ⊢ (𝜑 → ∪ {𝑡 ∣ {𝑥 ∣ 𝜓} = {𝑡}} = ∪ {𝑡 ∣ {𝑦 ∣ 𝜒} = {𝑡}})
6		df-iota 6520	. 2 ⊢ (℩𝑥𝜓) = ∪ {𝑡 ∣ {𝑥 ∣ 𝜓} = {𝑡}}
7		df-iota 6520	. 2 ⊢ (℩𝑦𝜒) = ∪ {𝑡 ∣ {𝑦 ∣ 𝜒} = {𝑡}}
8	5, 6, 7	3eqtr4g 2823	1 ⊢ (𝜑 → (℩𝑥𝜓) = (℩𝑦𝜒))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   = wceq 1570  {cab 2741  {csn 4588  ∪ cuni 4872  ℩cio 6518

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-ext 2735

This proof depends on definitions:  df-bi 210  df-an 402  df-tru 1573  df-ex 1813  df-sb 2102  df-clab 2742  df-cleq 2755  df-clel 2839  df-v 3456  df-ss 3901  df-uni 4873  df-iota 6520

This theorem is used by:  cbvriotadavw  37274  cbvriotadavw2  37294