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Theorem csbiota 6101
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 3742 . . . 4 (𝑧 = 𝐴𝑧 / 𝑥(℩𝑦𝜑) = 𝐴 / 𝑥(℩𝑦𝜑))
2 dfsbcq2 3647 . . . . 5 (𝑧 = 𝐴 → ([𝑧 / 𝑥]𝜑[𝐴 / 𝑥]𝜑))
32iotabidv 6092 . . . 4 (𝑧 = 𝐴 → (℩𝑦[𝑧 / 𝑥]𝜑) = (℩𝑦[𝐴 / 𝑥]𝜑))
41, 3eqeq12d 2832 . . 3 (𝑧 = 𝐴 → (𝑧 / 𝑥(℩𝑦𝜑) = (℩𝑦[𝑧 / 𝑥]𝜑) ↔ 𝐴 / 𝑥(℩𝑦𝜑) = (℩𝑦[𝐴 / 𝑥]𝜑)))
5 vex 3405 . . . 4 𝑧 ∈ V
6 nfs1v 2288 . . . . 5 𝑥[𝑧 / 𝑥]𝜑
76nfiota 6075 . . . 4 𝑥(℩𝑦[𝑧 / 𝑥]𝜑)
8 sbequ12 2280 . . . . 5 (𝑥 = 𝑧 → (𝜑 ↔ [𝑧 / 𝑥]𝜑))
98iotabidv 6092 . . . 4 (𝑥 = 𝑧 → (℩𝑦𝜑) = (℩𝑦[𝑧 / 𝑥]𝜑))
105, 7, 9csbief 3764 . . 3 𝑧 / 𝑥(℩𝑦𝜑) = (℩𝑦[𝑧 / 𝑥]𝜑)
114, 10vtoclg 3470 . 2 (𝐴 ∈ V → 𝐴 / 𝑥(℩𝑦𝜑) = (℩𝑦[𝐴 / 𝑥]𝜑))
12 csbprc 4189 . . 3 𝐴 ∈ V → 𝐴 / 𝑥(℩𝑦𝜑) = ∅)
13 sbcex 3654 . . . . . 6 ([𝐴 / 𝑥]𝜑𝐴 ∈ V)
1413con3i 151 . . . . 5 𝐴 ∈ V → ¬ [𝐴 / 𝑥]𝜑)
1514nexdv 2027 . . . 4 𝐴 ∈ V → ¬ ∃𝑦[𝐴 / 𝑥]𝜑)
16 euex 2667 . . . . 5 (∃!𝑦[𝐴 / 𝑥]𝜑 → ∃𝑦[𝐴 / 𝑥]𝜑)
1716con3i 151 . . . 4 (¬ ∃𝑦[𝐴 / 𝑥]𝜑 → ¬ ∃!𝑦[𝐴 / 𝑥]𝜑)
18 iotanul 6086 . . . 4 (¬ ∃!𝑦[𝐴 / 𝑥]𝜑 → (℩𝑦[𝐴 / 𝑥]𝜑) = ∅)
1915, 17, 183syl 18 . . 3 𝐴 ∈ V → (℩𝑦[𝐴 / 𝑥]𝜑) = ∅)
2012, 19eqtr4d 2854 . 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 1637  wex 1859  [wsb 2061  wcel 2157  ∃!weu 2641  Vcvv 3402  [wsbc 3644  csb 3739  c0 4127  cio 6069
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1877  ax-4 1894  ax-5 2001  ax-6 2069  ax-7 2105  ax-9 2166  ax-10 2186  ax-11 2202  ax-12 2215  ax-13 2422  ax-ext 2795
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 866  df-3an 1102  df-tru 1641  df-fal 1651  df-ex 1860  df-nf 1864  df-sb 2062  df-mo 2635  df-eu 2642  df-clab 2804  df-cleq 2810  df-clel 2813  df-nfc 2948  df-ral 3112  df-rex 3113  df-v 3404  df-sbc 3645  df-csb 3740  df-dif 3783  df-in 3787  df-ss 3794  df-nul 4128  df-sn 4382  df-uni 4642  df-iota 6071
This theorem is referenced by:  csbfv12  6458
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