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Theorem cbvriotavw 7398
Description: Change bound variable in a restricted description binder. Version of cbvriotav 7402 with a disjoint variable condition, which requires fewer axioms . (Contributed by NM, 18-Mar-2013.) (Revised by GG, 30-Sep-2024.)
Hypothesis
Ref Expression
cbvriotavw.1 (𝑥 = 𝑦 → (𝜑𝜓))
Assertion
Ref Expression
cbvriotavw (𝑥𝐴 𝜑) = (𝑦𝐴 𝜓)
Distinct variable groups:   𝑥,𝑦,𝐴   𝜑,𝑦   𝜓,𝑥
Allowed substitution hints:   𝜑(𝑥)   𝜓(𝑦)

Proof of Theorem cbvriotavw
StepHypRef Expression
1 eleq1w 2822 . . . 4 (𝑥 = 𝑦 → (𝑥𝐴𝑦𝐴))
2 cbvriotavw.1 . . . 4 (𝑥 = 𝑦 → (𝜑𝜓))
31, 2anbi12d 632 . . 3 (𝑥 = 𝑦 → ((𝑥𝐴𝜑) ↔ (𝑦𝐴𝜓)))
43cbviotavw 6524 . 2 (℩𝑥(𝑥𝐴𝜑)) = (℩𝑦(𝑦𝐴𝜓))
5 df-riota 7388 . 2 (𝑥𝐴 𝜑) = (℩𝑥(𝑥𝐴𝜑))
6 df-riota 7388 . 2 (𝑦𝐴 𝜓) = (℩𝑦(𝑦𝐴𝜓))
74, 5, 63eqtr4i 2773 1 (𝑥𝐴 𝜑) = (𝑦𝐴 𝜓)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395   = wceq 1537  wcel 2106  cio 6514  crio 7387
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-ext 2706
This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1540  df-ex 1777  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-v 3480  df-ss 3980  df-uni 4913  df-iota 6516  df-riota 7388
This theorem is referenced by:  ordtypecbv  9555  fin23lem27  10366  zorn2g  10541  nosupcbv  27762  noinfcbv  27777  uspgredg2v  29256  usgredg2v  29259  cnlnadji  32105  nmopadjlei  32117  cvmliftlem15  35283  cvmliftiota  35286  cvmlift2  35301  cvmlift3lem7  35310  cvmlift3  35313  weiunlem2  36446  lshpkrlem3  39094  cdleme40v  40452  lcfl7N  41484  lcf1o  41534  lcfrlem39  41564  hdmap1cbv  41785  wessf1ornlem  45128  fourierdlem103  46165  fourierdlem104  46166
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