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Theorem cbviotadavw 36212
Description: Change bound variable in a description binder. Deduction form. (Contributed by GG, 14-Aug-2025.)
Hypothesis
Ref Expression
cbviotadavw.1 ((𝜑𝑥 = 𝑦) → (𝜓𝜒))
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.
StepHypRef Expression
1 cbviotadavw.1 . . . . . 6 ((𝜑𝑥 = 𝑦) → (𝜓𝜒))
21cbvabdavw 36199 . . . . 5 (𝜑 → {𝑥𝜓} = {𝑦𝜒})
32eqeq1d 2735 . . . 4 (𝜑 → ({𝑥𝜓} = {𝑡} ↔ {𝑦𝜒} = {𝑡}))
43abbidv 2804 . . 3 (𝜑 → {𝑡 ∣ {𝑥𝜓} = {𝑡}} = {𝑡 ∣ {𝑦𝜒} = {𝑡}})
54unieqd 4927 . 2 (𝜑 {𝑡 ∣ {𝑥𝜓} = {𝑡}} = {𝑡 ∣ {𝑦𝜒} = {𝑡}})
6 df-iota 6510 . 2 (℩𝑥𝜓) = {𝑡 ∣ {𝑥𝜓} = {𝑡}}
7 df-iota 6510 . 2 (℩𝑦𝜒) = {𝑡 ∣ {𝑦𝜒} = {𝑡}}
85, 6, 73eqtr4g 2798 1 (𝜑 → (℩𝑥𝜓) = (℩𝑦𝜒))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395   = wceq 1535  {cab 2710  {csn 4630   cuni 4914  cio 6508
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1963  ax-7 2003  ax-8 2106  ax-9 2114  ax-ext 2704
This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1538  df-ex 1775  df-sb 2061  df-clab 2711  df-cleq 2725  df-clel 2812  df-v 3479  df-ss 3980  df-uni 4915  df-iota 6510
This theorem is referenced by:  cbvriotadavw  36213  cbvriotadavw2  36233
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