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Theorem csbiota 5919
Description: Class substitution within a description binder. (Contributed by Scott Fenton, 6-Oct-2017.) (Revised by NM, 23-Aug-2018.)
Assertion
Ref Expression
csbiota 𝐴 / 𝑥(℩𝑦𝜑) = (℩𝑦[𝐴 / 𝑥]𝜑)
Distinct variable groups:   𝑦,𝐴   𝑥,𝑦
Allowed substitution hints:   𝜑(𝑥,𝑦)   𝐴(𝑥)

Proof of Theorem csbiota
Dummy variable 𝑧 is distinct from all other variables.
StepHypRef Expression
1 csbeq1 3569 . . . 4 (𝑧 = 𝐴𝑧 / 𝑥(℩𝑦𝜑) = 𝐴 / 𝑥(℩𝑦𝜑))
2 dfsbcq2 3471 . . . . 5 (𝑧 = 𝐴 → ([𝑧 / 𝑥]𝜑[𝐴 / 𝑥]𝜑))
32iotabidv 5910 . . . 4 (𝑧 = 𝐴 → (℩𝑦[𝑧 / 𝑥]𝜑) = (℩𝑦[𝐴 / 𝑥]𝜑))
41, 3eqeq12d 2666 . . 3 (𝑧 = 𝐴 → (𝑧 / 𝑥(℩𝑦𝜑) = (℩𝑦[𝑧 / 𝑥]𝜑) ↔ 𝐴 / 𝑥(℩𝑦𝜑) = (℩𝑦[𝐴 / 𝑥]𝜑)))
5 vex 3234 . . . 4 𝑧 ∈ V
6 nfs1v 2465 . . . . 5 𝑥[𝑧 / 𝑥]𝜑
76nfiota 5893 . . . 4 𝑥(℩𝑦[𝑧 / 𝑥]𝜑)
8 sbequ12 2149 . . . . 5 (𝑥 = 𝑧 → (𝜑 ↔ [𝑧 / 𝑥]𝜑))
98iotabidv 5910 . . . 4 (𝑥 = 𝑧 → (℩𝑦𝜑) = (℩𝑦[𝑧 / 𝑥]𝜑))
105, 7, 9csbief 3591 . . 3 𝑧 / 𝑥(℩𝑦𝜑) = (℩𝑦[𝑧 / 𝑥]𝜑)
114, 10vtoclg 3297 . 2 (𝐴 ∈ V → 𝐴 / 𝑥(℩𝑦𝜑) = (℩𝑦[𝐴 / 𝑥]𝜑))
12 csbprc 4013 . . 3 𝐴 ∈ V → 𝐴 / 𝑥(℩𝑦𝜑) = ∅)
13 sbcex 3478 . . . . . 6 ([𝐴 / 𝑥]𝜑𝐴 ∈ V)
1413con3i 150 . . . . 5 𝐴 ∈ V → ¬ [𝐴 / 𝑥]𝜑)
1514nexdv 1904 . . . 4 𝐴 ∈ V → ¬ ∃𝑦[𝐴 / 𝑥]𝜑)
16 euex 2522 . . . . 5 (∃!𝑦[𝐴 / 𝑥]𝜑 → ∃𝑦[𝐴 / 𝑥]𝜑)
1716con3i 150 . . . 4 (¬ ∃𝑦[𝐴 / 𝑥]𝜑 → ¬ ∃!𝑦[𝐴 / 𝑥]𝜑)
18 iotanul 5904 . . . 4 (¬ ∃!𝑦[𝐴 / 𝑥]𝜑 → (℩𝑦[𝐴 / 𝑥]𝜑) = ∅)
1915, 17, 183syl 18 . . 3 𝐴 ∈ V → (℩𝑦[𝐴 / 𝑥]𝜑) = ∅)
2012, 19eqtr4d 2688 . 2 𝐴 ∈ V → 𝐴 / 𝑥(℩𝑦𝜑) = (℩𝑦[𝐴 / 𝑥]𝜑))
2111, 20pm2.61i 176 1 𝐴 / 𝑥(℩𝑦𝜑) = (℩𝑦[𝐴 / 𝑥]𝜑)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3   = wceq 1523  wex 1744  [wsb 1937  wcel 2030  ∃!weu 2498  Vcvv 3231  [wsbc 3468  csb 3566  c0 3948  cio 5887
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-fal 1529  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ral 2946  df-rex 2947  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-in 3614  df-ss 3621  df-nul 3949  df-sn 4211  df-uni 4469  df-iota 5889
This theorem is referenced by:  csbfv12  6269
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