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Theorem cbvriotavw 7318
Description: Change bound variable in a restricted description binder. Version of cbvriotav 7323 with a disjoint variable condition, which requires fewer axioms . (Contributed by NM, 18-Mar-2013.) (Revised by Gino Giotto, 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 2821 . . . 4 (𝑥 = 𝑦 → (𝑥𝐴𝑦𝐴))
2 cbvriotavw.1 . . . 4 (𝑥 = 𝑦 → (𝜑𝜓))
31, 2anbi12d 632 . . 3 (𝑥 = 𝑦 → ((𝑥𝐴𝜑) ↔ (𝑦𝐴𝜓)))
43cbviotavw 6454 . 2 (℩𝑥(𝑥𝐴𝜑)) = (℩𝑦(𝑦𝐴𝜓))
5 df-riota 7308 . 2 (𝑥𝐴 𝜑) = (℩𝑥(𝑥𝐴𝜑))
6 df-riota 7308 . 2 (𝑦𝐴 𝜓) = (℩𝑦(𝑦𝐴𝜓))
74, 5, 63eqtr4i 2776 1 (𝑥𝐴 𝜑) = (𝑦𝐴 𝜓)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 397   = wceq 1542  wcel 2107  cio 6444  crio 7307
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-ext 2709
This theorem depends on definitions:  df-bi 206  df-an 398  df-tru 1545  df-ex 1783  df-sb 2069  df-clab 2716  df-cleq 2730  df-clel 2816  df-v 3446  df-in 3916  df-ss 3926  df-uni 4865  df-iota 6446  df-riota 7308
This theorem is referenced by:  ordtypecbv  9412  fin23lem27  10223  zorn2g  10398  nosupcbv  27002  noinfcbv  27017  uspgredg2v  28001  usgredg2v  28004  cnlnadji  30847  nmopadjlei  30859  cvmliftlem15  33696  cvmliftiota  33699  cvmlift2  33714  cvmlift3lem7  33723  cvmlift3  33726  lshpkrlem3  37506  cdleme40v  38864  lcfl7N  39896  lcf1o  39946  lcfrlem39  39976  hdmap1cbv  40197  wessf1ornlem  43302  fourierdlem103  44345  fourierdlem104  44346
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