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Theorem cbviotavw 6502
Description: Change bound variables in a description binder. Version of cbviotav 6505 with a disjoint variable condition, which requires fewer axioms . (Contributed by Andrew Salmon, 1-Aug-2011.) (Revised by Gino Giotto, 30-Sep-2024.)
Hypothesis
Ref Expression
cbviotavw.1 (𝑥 = 𝑦 → (𝜑𝜓))
Assertion
Ref Expression
cbviotavw (℩𝑥𝜑) = (℩𝑦𝜓)
Distinct variable groups:   𝜑,𝑦   𝜓,𝑥   𝑥,𝑦
Allowed substitution hints:   𝜑(𝑥)   𝜓(𝑦)

Proof of Theorem cbviotavw
Dummy variable 𝑧 is distinct from all other variables.
StepHypRef Expression
1 cbviotavw.1 . . . . . 6 (𝑥 = 𝑦 → (𝜑𝜓))
21cbvabv 2803 . . . . 5 {𝑥𝜑} = {𝑦𝜓}
32eqeq1i 2735 . . . 4 ({𝑥𝜑} = {𝑧} ↔ {𝑦𝜓} = {𝑧})
43abbii 2800 . . 3 {𝑧 ∣ {𝑥𝜑} = {𝑧}} = {𝑧 ∣ {𝑦𝜓} = {𝑧}}
54unieqi 4920 . 2 {𝑧 ∣ {𝑥𝜑} = {𝑧}} = {𝑧 ∣ {𝑦𝜓} = {𝑧}}
6 df-iota 6494 . 2 (℩𝑥𝜑) = {𝑧 ∣ {𝑥𝜑} = {𝑧}}
7 df-iota 6494 . 2 (℩𝑦𝜓) = {𝑧 ∣ {𝑦𝜓} = {𝑧}}
85, 6, 73eqtr4i 2768 1 (℩𝑥𝜑) = (℩𝑦𝜓)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205   = wceq 1539  {cab 2707  {csn 4627   cuni 4907  cio 6492
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1911  ax-6 1969  ax-7 2009  ax-8 2106  ax-9 2114  ax-ext 2701
This theorem depends on definitions:  df-bi 206  df-an 395  df-tru 1542  df-ex 1780  df-sb 2066  df-clab 2708  df-cleq 2722  df-clel 2808  df-v 3474  df-in 3954  df-ss 3964  df-uni 4908  df-iota 6494
This theorem is referenced by:  cbvriotavw  7377  oeeui  8604  nosupcbv  27441  noinfcbv  27456  ellimciota  44628  fourierdlem96  45216  fourierdlem97  45217  fourierdlem98  45218  fourierdlem99  45219  fourierdlem105  45225  fourierdlem106  45226  fourierdlem108  45228  fourierdlem110  45230  fourierdlem112  45232  fourierdlem113  45233  fourierdlem115  45235  funressndmafv2rn  46229
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