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Theorem csbiota 6023
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 3685 . . . 4 (𝑧 = 𝐴𝑧 / 𝑥(℩𝑦𝜑) = 𝐴 / 𝑥(℩𝑦𝜑))
2 dfsbcq2 3590 . . . . 5 (𝑧 = 𝐴 → ([𝑧 / 𝑥]𝜑[𝐴 / 𝑥]𝜑))
32iotabidv 6014 . . . 4 (𝑧 = 𝐴 → (℩𝑦[𝑧 / 𝑥]𝜑) = (℩𝑦[𝐴 / 𝑥]𝜑))
41, 3eqeq12d 2786 . . 3 (𝑧 = 𝐴 → (𝑧 / 𝑥(℩𝑦𝜑) = (℩𝑦[𝑧 / 𝑥]𝜑) ↔ 𝐴 / 𝑥(℩𝑦𝜑) = (℩𝑦[𝐴 / 𝑥]𝜑)))
5 vex 3354 . . . 4 𝑧 ∈ V
6 nfs1v 2274 . . . . 5 𝑥[𝑧 / 𝑥]𝜑
76nfiota 5997 . . . 4 𝑥(℩𝑦[𝑧 / 𝑥]𝜑)
8 sbequ12 2267 . . . . 5 (𝑥 = 𝑧 → (𝜑 ↔ [𝑧 / 𝑥]𝜑))
98iotabidv 6014 . . . 4 (𝑥 = 𝑧 → (℩𝑦𝜑) = (℩𝑦[𝑧 / 𝑥]𝜑))
105, 7, 9csbief 3707 . . 3 𝑧 / 𝑥(℩𝑦𝜑) = (℩𝑦[𝑧 / 𝑥]𝜑)
114, 10vtoclg 3417 . 2 (𝐴 ∈ V → 𝐴 / 𝑥(℩𝑦𝜑) = (℩𝑦[𝐴 / 𝑥]𝜑))
12 csbprc 4125 . . 3 𝐴 ∈ V → 𝐴 / 𝑥(℩𝑦𝜑) = ∅)
13 sbcex 3597 . . . . . 6 ([𝐴 / 𝑥]𝜑𝐴 ∈ V)
1413con3i 151 . . . . 5 𝐴 ∈ V → ¬ [𝐴 / 𝑥]𝜑)
1514nexdv 2016 . . . 4 𝐴 ∈ V → ¬ ∃𝑦[𝐴 / 𝑥]𝜑)
16 euex 2642 . . . . 5 (∃!𝑦[𝐴 / 𝑥]𝜑 → ∃𝑦[𝐴 / 𝑥]𝜑)
1716con3i 151 . . . 4 (¬ ∃𝑦[𝐴 / 𝑥]𝜑 → ¬ ∃!𝑦[𝐴 / 𝑥]𝜑)
18 iotanul 6008 . . . 4 (¬ ∃!𝑦[𝐴 / 𝑥]𝜑 → (℩𝑦[𝐴 / 𝑥]𝜑) = ∅)
1915, 17, 183syl 18 . . 3 𝐴 ∈ V → (℩𝑦[𝐴 / 𝑥]𝜑) = ∅)
2012, 19eqtr4d 2808 . 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 1631  wex 1852  [wsb 2049  wcel 2145  ∃!weu 2618  Vcvv 3351  [wsbc 3587  csb 3682  c0 4063  cio 5991
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-3an 1073  df-tru 1634  df-fal 1637  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ral 3066  df-rex 3067  df-v 3353  df-sbc 3588  df-csb 3683  df-dif 3726  df-in 3730  df-ss 3737  df-nul 4064  df-sn 4318  df-uni 4576  df-iota 5993
This theorem is referenced by:  csbfv12  6374
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