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Theorem csbriota 7118
Description: Interchange class substitution and restricted description binder. (Contributed by NM, 24-Feb-2013.) (Revised by NM, 2-Sep-2018.)
Assertion
Ref Expression
csbriota 𝐴 / 𝑥(𝑦𝐵 𝜑) = (𝑦𝐵 [𝐴 / 𝑥]𝜑)
Distinct variable groups:   𝑦,𝐴   𝑥,𝐵   𝑥,𝑦
Allowed substitution hints:   𝜑(𝑥,𝑦)   𝐴(𝑥)   𝐵(𝑦)

Proof of Theorem csbriota
Dummy variable 𝑧 is distinct from all other variables.
StepHypRef Expression
1 csbeq1 3883 . . . 4 (𝑧 = 𝐴𝑧 / 𝑥(𝑦𝐵 𝜑) = 𝐴 / 𝑥(𝑦𝐵 𝜑))
2 dfsbcq2 3772 . . . . 5 (𝑧 = 𝐴 → ([𝑧 / 𝑥]𝜑[𝐴 / 𝑥]𝜑))
32riotabidv 7105 . . . 4 (𝑧 = 𝐴 → (𝑦𝐵 [𝑧 / 𝑥]𝜑) = (𝑦𝐵 [𝐴 / 𝑥]𝜑))
41, 3eqeq12d 2834 . . 3 (𝑧 = 𝐴 → (𝑧 / 𝑥(𝑦𝐵 𝜑) = (𝑦𝐵 [𝑧 / 𝑥]𝜑) ↔ 𝐴 / 𝑥(𝑦𝐵 𝜑) = (𝑦𝐵 [𝐴 / 𝑥]𝜑)))
5 vex 3495 . . . 4 𝑧 ∈ V
6 nfs1v 2264 . . . . 5 𝑥[𝑧 / 𝑥]𝜑
7 nfcv 2974 . . . . 5 𝑥𝐵
86, 7nfriota 7115 . . . 4 𝑥(𝑦𝐵 [𝑧 / 𝑥]𝜑)
9 sbequ12 2243 . . . . 5 (𝑥 = 𝑧 → (𝜑 ↔ [𝑧 / 𝑥]𝜑))
109riotabidv 7105 . . . 4 (𝑥 = 𝑧 → (𝑦𝐵 𝜑) = (𝑦𝐵 [𝑧 / 𝑥]𝜑))
115, 8, 10csbief 3914 . . 3 𝑧 / 𝑥(𝑦𝐵 𝜑) = (𝑦𝐵 [𝑧 / 𝑥]𝜑)
124, 11vtoclg 3565 . 2 (𝐴 ∈ V → 𝐴 / 𝑥(𝑦𝐵 𝜑) = (𝑦𝐵 [𝐴 / 𝑥]𝜑))
13 csbprc 4355 . . 3 𝐴 ∈ V → 𝐴 / 𝑥(𝑦𝐵 𝜑) = ∅)
14 df-riota 7103 . . . 4 (𝑦𝐵 [𝐴 / 𝑥]𝜑) = (℩𝑦(𝑦𝐵[𝐴 / 𝑥]𝜑))
15 euex 2655 . . . . . . 7 (∃!𝑦(𝑦𝐵[𝐴 / 𝑥]𝜑) → ∃𝑦(𝑦𝐵[𝐴 / 𝑥]𝜑))
16 sbcex 3779 . . . . . . . . 9 ([𝐴 / 𝑥]𝜑𝐴 ∈ V)
1716adantl 482 . . . . . . . 8 ((𝑦𝐵[𝐴 / 𝑥]𝜑) → 𝐴 ∈ V)
1817exlimiv 1922 . . . . . . 7 (∃𝑦(𝑦𝐵[𝐴 / 𝑥]𝜑) → 𝐴 ∈ V)
1915, 18syl 17 . . . . . 6 (∃!𝑦(𝑦𝐵[𝐴 / 𝑥]𝜑) → 𝐴 ∈ V)
2019con3i 157 . . . . 5 𝐴 ∈ V → ¬ ∃!𝑦(𝑦𝐵[𝐴 / 𝑥]𝜑))
21 iotanul 6326 . . . . 5 (¬ ∃!𝑦(𝑦𝐵[𝐴 / 𝑥]𝜑) → (℩𝑦(𝑦𝐵[𝐴 / 𝑥]𝜑)) = ∅)
2220, 21syl 17 . . . 4 𝐴 ∈ V → (℩𝑦(𝑦𝐵[𝐴 / 𝑥]𝜑)) = ∅)
2314, 22syl5req 2866 . . 3 𝐴 ∈ V → ∅ = (𝑦𝐵 [𝐴 / 𝑥]𝜑))
2413, 23eqtrd 2853 . 2 𝐴 ∈ V → 𝐴 / 𝑥(𝑦𝐵 𝜑) = (𝑦𝐵 [𝐴 / 𝑥]𝜑))
2512, 24pm2.61i 183 1 𝐴 / 𝑥(𝑦𝐵 𝜑) = (𝑦𝐵 [𝐴 / 𝑥]𝜑)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wa 396   = wceq 1528  wex 1771  [wsb 2060  wcel 2105  ∃!weu 2646  Vcvv 3492  [wsbc 3769  csb 3880  c0 4288  cio 6305  crio 7102
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3an 1081  df-tru 1531  df-fal 1541  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ral 3140  df-rex 3141  df-v 3494  df-sbc 3770  df-csb 3881  df-dif 3936  df-in 3940  df-ss 3949  df-nul 4289  df-sn 4558  df-uni 4831  df-iota 6307  df-riota 7103
This theorem is referenced by:  cdlemkid3N  37949  cdlemkid4  37950
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