Theorem cbviotavw 6528

Description: Change bound variables in a description binder. Version of cbviotav 6530 with a disjoint variable condition, which requires fewer axioms . (Contributed by Andrew Salmon, 1-Aug-2011.) (Revised by GG, 30-Sep-2024.)

Hypothesis

Ref	Expression
cbviotavw.1	⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓))

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.

Step	Hyp	Ref	Expression
1		cbviotavw.1	. . . . . 6 ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓))
2	1	cbvabv 2834	. . . . 5 ⊢ {𝑥 ∣ 𝜑} = {𝑦 ∣ 𝜓}
3	2	eqeq1i 2768	. . . 4 ⊢ ({𝑥 ∣ 𝜑} = {𝑧} ↔ {𝑦 ∣ 𝜓} = {𝑧})
4	3	abbii 2830	. . 3 ⊢ {𝑧 ∣ {𝑥 ∣ 𝜑} = {𝑧}} = {𝑧 ∣ {𝑦 ∣ 𝜓} = {𝑧}}
5	4	unieqi 4884	. 2 ⊢ ∪ {𝑧 ∣ {𝑥 ∣ 𝜑} = {𝑧}} = ∪ {𝑧 ∣ {𝑦 ∣ 𝜓} = {𝑧}}
6		df-iota 6520	. 2 ⊢ (℩𝑥𝜑) = ∪ {𝑧 ∣ {𝑥 ∣ 𝜑} = {𝑧}}
7		df-iota 6520	. 2 ⊢ (℩𝑦𝜓) = ∪ {𝑧 ∣ {𝑦 ∣ 𝜓} = {𝑧}}
8	5, 6, 7	3eqtr4i 2796	1 ⊢ (℩𝑥𝜑) = (℩𝑦𝜓)

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   = 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:  cbvriotavw  7418  oeeui  8649  nosupcbv  28116  noinfcbv  28131  cbvriotavw2  37240  ellimciota  46899  fourierdlem96  47485  fourierdlem97  47486  fourierdlem98  47487  fourierdlem99  47488  fourierdlem105  47494  fourierdlem106  47495  fourierdlem108  47497  fourierdlem110  47499  fourierdlem112  47501  fourierdlem113  47502  fourierdlem115  47504  funressndmafv2rn  48566