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Theorem csbriota 6663
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 3569 . . . 4 (𝑧 = 𝐴𝑧 / 𝑥(𝑦𝐵 𝜑) = 𝐴 / 𝑥(𝑦𝐵 𝜑))
2 dfsbcq2 3471 . . . . 5 (𝑧 = 𝐴 → ([𝑧 / 𝑥]𝜑[𝐴 / 𝑥]𝜑))
32riotabidv 6653 . . . 4 (𝑧 = 𝐴 → (𝑦𝐵 [𝑧 / 𝑥]𝜑) = (𝑦𝐵 [𝐴 / 𝑥]𝜑))
41, 3eqeq12d 2666 . . 3 (𝑧 = 𝐴 → (𝑧 / 𝑥(𝑦𝐵 𝜑) = (𝑦𝐵 [𝑧 / 𝑥]𝜑) ↔ 𝐴 / 𝑥(𝑦𝐵 𝜑) = (𝑦𝐵 [𝐴 / 𝑥]𝜑)))
5 vex 3234 . . . 4 𝑧 ∈ V
6 nfs1v 2465 . . . . 5 𝑥[𝑧 / 𝑥]𝜑
7 nfcv 2793 . . . . 5 𝑥𝐵
86, 7nfriota 6660 . . . 4 𝑥(𝑦𝐵 [𝑧 / 𝑥]𝜑)
9 sbequ12 2149 . . . . 5 (𝑥 = 𝑧 → (𝜑 ↔ [𝑧 / 𝑥]𝜑))
109riotabidv 6653 . . . 4 (𝑥 = 𝑧 → (𝑦𝐵 𝜑) = (𝑦𝐵 [𝑧 / 𝑥]𝜑))
115, 8, 10csbief 3591 . . 3 𝑧 / 𝑥(𝑦𝐵 𝜑) = (𝑦𝐵 [𝑧 / 𝑥]𝜑)
124, 11vtoclg 3297 . 2 (𝐴 ∈ V → 𝐴 / 𝑥(𝑦𝐵 𝜑) = (𝑦𝐵 [𝐴 / 𝑥]𝜑))
13 csbprc 4013 . . 3 𝐴 ∈ V → 𝐴 / 𝑥(𝑦𝐵 𝜑) = ∅)
14 df-riota 6651 . . . 4 (𝑦𝐵 [𝐴 / 𝑥]𝜑) = (℩𝑦(𝑦𝐵[𝐴 / 𝑥]𝜑))
15 euex 2522 . . . . . . 7 (∃!𝑦(𝑦𝐵[𝐴 / 𝑥]𝜑) → ∃𝑦(𝑦𝐵[𝐴 / 𝑥]𝜑))
16 sbcex 3478 . . . . . . . . 9 ([𝐴 / 𝑥]𝜑𝐴 ∈ V)
1716adantl 481 . . . . . . . 8 ((𝑦𝐵[𝐴 / 𝑥]𝜑) → 𝐴 ∈ V)
1817exlimiv 1898 . . . . . . 7 (∃𝑦(𝑦𝐵[𝐴 / 𝑥]𝜑) → 𝐴 ∈ V)
1915, 18syl 17 . . . . . 6 (∃!𝑦(𝑦𝐵[𝐴 / 𝑥]𝜑) → 𝐴 ∈ V)
2019con3i 150 . . . . 5 𝐴 ∈ V → ¬ ∃!𝑦(𝑦𝐵[𝐴 / 𝑥]𝜑))
21 iotanul 5904 . . . . 5 (¬ ∃!𝑦(𝑦𝐵[𝐴 / 𝑥]𝜑) → (℩𝑦(𝑦𝐵[𝐴 / 𝑥]𝜑)) = ∅)
2220, 21syl 17 . . . 4 𝐴 ∈ V → (℩𝑦(𝑦𝐵[𝐴 / 𝑥]𝜑)) = ∅)
2314, 22syl5req 2698 . . 3 𝐴 ∈ V → ∅ = (𝑦𝐵 [𝐴 / 𝑥]𝜑))
2413, 23eqtrd 2685 . 2 𝐴 ∈ V → 𝐴 / 𝑥(𝑦𝐵 𝜑) = (𝑦𝐵 [𝐴 / 𝑥]𝜑))
2512, 24pm2.61i 176 1 𝐴 / 𝑥(𝑦𝐵 𝜑) = (𝑦𝐵 [𝐴 / 𝑥]𝜑)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wa 383   = wceq 1523  wex 1744  [wsb 1937  wcel 2030  ∃!weu 2498  Vcvv 3231  [wsbc 3468  csb 3566  c0 3948  cio 5887  crio 6650
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  df-riota 6651
This theorem is referenced by:  cdlemkid3N  36538  cdlemkid4  36539
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