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Theorem csbiota 6556
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 3911 . . . 4 (𝑧 = 𝐴𝑧 / 𝑥(℩𝑦𝜑) = 𝐴 / 𝑥(℩𝑦𝜑))
2 dfsbcq2 3794 . . . . 5 (𝑧 = 𝐴 → ([𝑧 / 𝑥]𝜑[𝐴 / 𝑥]𝜑))
32iotabidv 6547 . . . 4 (𝑧 = 𝐴 → (℩𝑦[𝑧 / 𝑥]𝜑) = (℩𝑦[𝐴 / 𝑥]𝜑))
41, 3eqeq12d 2751 . . 3 (𝑧 = 𝐴 → (𝑧 / 𝑥(℩𝑦𝜑) = (℩𝑦[𝑧 / 𝑥]𝜑) ↔ 𝐴 / 𝑥(℩𝑦𝜑) = (℩𝑦[𝐴 / 𝑥]𝜑)))
5 vex 3482 . . . 4 𝑧 ∈ V
6 nfs1v 2154 . . . . 5 𝑥[𝑧 / 𝑥]𝜑
76nfiotaw 6520 . . . 4 𝑥(℩𝑦[𝑧 / 𝑥]𝜑)
8 sbequ12 2249 . . . . 5 (𝑥 = 𝑧 → (𝜑 ↔ [𝑧 / 𝑥]𝜑))
98iotabidv 6547 . . . 4 (𝑥 = 𝑧 → (℩𝑦𝜑) = (℩𝑦[𝑧 / 𝑥]𝜑))
105, 7, 9csbief 3943 . . 3 𝑧 / 𝑥(℩𝑦𝜑) = (℩𝑦[𝑧 / 𝑥]𝜑)
114, 10vtoclg 3554 . 2 (𝐴 ∈ V → 𝐴 / 𝑥(℩𝑦𝜑) = (℩𝑦[𝐴 / 𝑥]𝜑))
12 csbprc 4415 . . 3 𝐴 ∈ V → 𝐴 / 𝑥(℩𝑦𝜑) = ∅)
13 sbcex 3801 . . . . . 6 ([𝐴 / 𝑥]𝜑𝐴 ∈ V)
1413con3i 154 . . . . 5 𝐴 ∈ V → ¬ [𝐴 / 𝑥]𝜑)
1514nexdv 1934 . . . 4 𝐴 ∈ V → ¬ ∃𝑦[𝐴 / 𝑥]𝜑)
16 euex 2575 . . . . 5 (∃!𝑦[𝐴 / 𝑥]𝜑 → ∃𝑦[𝐴 / 𝑥]𝜑)
1716con3i 154 . . . 4 (¬ ∃𝑦[𝐴 / 𝑥]𝜑 → ¬ ∃!𝑦[𝐴 / 𝑥]𝜑)
18 iotanul 6541 . . . 4 (¬ ∃!𝑦[𝐴 / 𝑥]𝜑 → (℩𝑦[𝐴 / 𝑥]𝜑) = ∅)
1915, 17, 183syl 18 . . 3 𝐴 ∈ V → (℩𝑦[𝐴 / 𝑥]𝜑) = ∅)
2012, 19eqtr4d 2778 . 2 𝐴 ∈ V → 𝐴 / 𝑥(℩𝑦𝜑) = (℩𝑦[𝐴 / 𝑥]𝜑))
2111, 20pm2.61i 182 1 𝐴 / 𝑥(℩𝑦𝜑) = (℩𝑦[𝐴 / 𝑥]𝜑)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3   = wceq 1537  wex 1776  [wsb 2062  wcel 2106  ∃!weu 2566  Vcvv 3478  [wsbc 3791  csb 3908  c0 4339  cio 6514
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-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ral 3060  df-rex 3069  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-ss 3980  df-nul 4340  df-sn 4632  df-uni 4913  df-iota 6516
This theorem is referenced by:  csbfv12  6955
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