Users' Mathboxes Mathbox for Alan Sare < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  en3lplem1VD Structured version   Visualization version   GIF version

Theorem en3lplem1VD 39392
Description: Virtual deduction proof of en3lplem1 8549. (Contributed by Alan Sare, 24-Oct-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
Assertion
Ref Expression
en3lplem1VD ((𝐴𝐵𝐵𝐶𝐶𝐴) → (𝑥 = 𝐴 → ∃𝑦(𝑦 ∈ {𝐴, 𝐵, 𝐶} ∧ 𝑦𝑥)))
Distinct variable groups:   𝑥,𝑦,𝐴   𝑥,𝐵,𝑦   𝑥,𝐶,𝑦

Proof of Theorem en3lplem1VD
StepHypRef Expression
1 idn1 39107 . . . . . . 7 (   (𝐴𝐵𝐵𝐶𝐶𝐴)   ▶   (𝐴𝐵𝐵𝐶𝐶𝐴)   )
2 simp3 1083 . . . . . . 7 ((𝐴𝐵𝐵𝐶𝐶𝐴) → 𝐶𝐴)
31, 2e1a 39169 . . . . . 6 (   (𝐴𝐵𝐵𝐶𝐶𝐴)   ▶   𝐶𝐴   )
4 tpid3g 4337 . . . . . 6 (𝐶𝐴𝐶 ∈ {𝐴, 𝐵, 𝐶})
53, 4e1a 39169 . . . . 5 (   (𝐴𝐵𝐵𝐶𝐶𝐴)   ▶   𝐶 ∈ {𝐴, 𝐵, 𝐶}   )
6 idn2 39155 . . . . . 6 (   (𝐴𝐵𝐵𝐶𝐶𝐴)   ,   𝑥 = 𝐴   ▶   𝑥 = 𝐴   )
7 eleq2 2719 . . . . . . 7 (𝑥 = 𝐴 → (𝐶𝑥𝐶𝐴))
87biimprd 238 . . . . . 6 (𝑥 = 𝐴 → (𝐶𝐴𝐶𝑥))
96, 3, 8e21 39274 . . . . 5 (   (𝐴𝐵𝐵𝐶𝐶𝐴)   ,   𝑥 = 𝐴   ▶   𝐶𝑥   )
10 pm3.2 462 . . . . 5 (𝐶 ∈ {𝐴, 𝐵, 𝐶} → (𝐶𝑥 → (𝐶 ∈ {𝐴, 𝐵, 𝐶} ∧ 𝐶𝑥)))
115, 9, 10e12 39268 . . . 4 (   (𝐴𝐵𝐵𝐶𝐶𝐴)   ,   𝑥 = 𝐴   ▶   (𝐶 ∈ {𝐴, 𝐵, 𝐶} ∧ 𝐶𝑥)   )
12 elex22 3248 . . . 4 ((𝐶 ∈ {𝐴, 𝐵, 𝐶} ∧ 𝐶𝑥) → ∃𝑦(𝑦 ∈ {𝐴, 𝐵, 𝐶} ∧ 𝑦𝑥))
1311, 12e2 39173 . . 3 (   (𝐴𝐵𝐵𝐶𝐶𝐴)   ,   𝑥 = 𝐴   ▶   𝑦(𝑦 ∈ {𝐴, 𝐵, 𝐶} ∧ 𝑦𝑥)   )
1413in2 39147 . 2 (   (𝐴𝐵𝐵𝐶𝐶𝐴)   ▶   (𝑥 = 𝐴 → ∃𝑦(𝑦 ∈ {𝐴, 𝐵, 𝐶} ∧ 𝑦𝑥))   )
1514in1 39104 1 ((𝐴𝐵𝐵𝐶𝐶𝐴) → (𝑥 = 𝐴 → ∃𝑦(𝑦 ∈ {𝐴, 𝐵, 𝐶} ∧ 𝑦𝑥)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 383  w3a 1054   = wceq 1523  wex 1744  wcel 2030  {ctp 4214
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-3or 1055  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-v 3233  df-un 3612  df-sn 4211  df-pr 4213  df-tp 4215  df-vd1 39103  df-vd2 39111
This theorem is referenced by:  en3lplem2VD  39393
  Copyright terms: Public domain W3C validator