MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  csbiota Structured version   Visualization version   GIF version

Theorem csbiota 6554
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 3902 . . . 4 (𝑧 = 𝐴𝑧 / 𝑥(℩𝑦𝜑) = 𝐴 / 𝑥(℩𝑦𝜑))
2 dfsbcq2 3791 . . . . 5 (𝑧 = 𝐴 → ([𝑧 / 𝑥]𝜑[𝐴 / 𝑥]𝜑))
32iotabidv 6545 . . . 4 (𝑧 = 𝐴 → (℩𝑦[𝑧 / 𝑥]𝜑) = (℩𝑦[𝐴 / 𝑥]𝜑))
41, 3eqeq12d 2753 . . 3 (𝑧 = 𝐴 → (𝑧 / 𝑥(℩𝑦𝜑) = (℩𝑦[𝑧 / 𝑥]𝜑) ↔ 𝐴 / 𝑥(℩𝑦𝜑) = (℩𝑦[𝐴 / 𝑥]𝜑)))
5 vex 3484 . . . 4 𝑧 ∈ V
6 nfs1v 2156 . . . . 5 𝑥[𝑧 / 𝑥]𝜑
76nfiotaw 6518 . . . 4 𝑥(℩𝑦[𝑧 / 𝑥]𝜑)
8 sbequ12 2251 . . . . 5 (𝑥 = 𝑧 → (𝜑 ↔ [𝑧 / 𝑥]𝜑))
98iotabidv 6545 . . . 4 (𝑥 = 𝑧 → (℩𝑦𝜑) = (℩𝑦[𝑧 / 𝑥]𝜑))
105, 7, 9csbief 3933 . . 3 𝑧 / 𝑥(℩𝑦𝜑) = (℩𝑦[𝑧 / 𝑥]𝜑)
114, 10vtoclg 3554 . 2 (𝐴 ∈ V → 𝐴 / 𝑥(℩𝑦𝜑) = (℩𝑦[𝐴 / 𝑥]𝜑))
12 csbprc 4409 . . 3 𝐴 ∈ V → 𝐴 / 𝑥(℩𝑦𝜑) = ∅)
13 sbcex 3798 . . . . . 6 ([𝐴 / 𝑥]𝜑𝐴 ∈ V)
1413con3i 154 . . . . 5 𝐴 ∈ V → ¬ [𝐴 / 𝑥]𝜑)
1514nexdv 1936 . . . 4 𝐴 ∈ V → ¬ ∃𝑦[𝐴 / 𝑥]𝜑)
16 euex 2577 . . . . 5 (∃!𝑦[𝐴 / 𝑥]𝜑 → ∃𝑦[𝐴 / 𝑥]𝜑)
1716con3i 154 . . . 4 (¬ ∃𝑦[𝐴 / 𝑥]𝜑 → ¬ ∃!𝑦[𝐴 / 𝑥]𝜑)
18 iotanul 6539 . . . 4 (¬ ∃!𝑦[𝐴 / 𝑥]𝜑 → (℩𝑦[𝐴 / 𝑥]𝜑) = ∅)
1915, 17, 183syl 18 . . 3 𝐴 ∈ V → (℩𝑦[𝐴 / 𝑥]𝜑) = ∅)
2012, 19eqtr4d 2780 . 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 1540  wex 1779  [wsb 2064  wcel 2108  ∃!weu 2568  Vcvv 3480  [wsbc 3788  csb 3899  c0 4333  cio 6512
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ral 3062  df-rex 3071  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-ss 3968  df-nul 4334  df-sn 4627  df-uni 4908  df-iota 6514
This theorem is referenced by:  csbfv12  6954
  Copyright terms: Public domain W3C validator