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Theorem en3lplem1VD 39577
Description: Virtual deduction proof of en3lplem1 8680. (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 39292 . . . . . . 7 (   (𝐴𝐵𝐵𝐶𝐶𝐴)   ▶   (𝐴𝐵𝐵𝐶𝐶𝐴)   )
2 simp3 1133 . . . . . . 7 ((𝐴𝐵𝐵𝐶𝐶𝐴) → 𝐶𝐴)
31, 2e1a 39354 . . . . . 6 (   (𝐴𝐵𝐵𝐶𝐶𝐴)   ▶   𝐶𝐴   )
4 tpid3g 4449 . . . . . 6 (𝐶𝐴𝐶 ∈ {𝐴, 𝐵, 𝐶})
53, 4e1a 39354 . . . . 5 (   (𝐴𝐵𝐵𝐶𝐶𝐴)   ▶   𝐶 ∈ {𝐴, 𝐵, 𝐶}   )
6 idn2 39340 . . . . . 6 (   (𝐴𝐵𝐵𝐶𝐶𝐴)   ,   𝑥 = 𝐴   ▶   𝑥 = 𝐴   )
7 eleq2 2828 . . . . . . 7 (𝑥 = 𝐴 → (𝐶𝑥𝐶𝐴))
87biimprd 238 . . . . . 6 (𝑥 = 𝐴 → (𝐶𝐴𝐶𝑥))
96, 3, 8e21 39459 . . . . 5 (   (𝐴𝐵𝐵𝐶𝐶𝐴)   ,   𝑥 = 𝐴   ▶   𝐶𝑥   )
10 pm3.2 462 . . . . 5 (𝐶 ∈ {𝐴, 𝐵, 𝐶} → (𝐶𝑥 → (𝐶 ∈ {𝐴, 𝐵, 𝐶} ∧ 𝐶𝑥)))
115, 9, 10e12 39453 . . . 4 (   (𝐴𝐵𝐵𝐶𝐶𝐴)   ,   𝑥 = 𝐴   ▶   (𝐶 ∈ {𝐴, 𝐵, 𝐶} ∧ 𝐶𝑥)   )
12 elex22 3357 . . . 4 ((𝐶 ∈ {𝐴, 𝐵, 𝐶} ∧ 𝐶𝑥) → ∃𝑦(𝑦 ∈ {𝐴, 𝐵, 𝐶} ∧ 𝑦𝑥))
1311, 12e2 39358 . . 3 (   (𝐴𝐵𝐵𝐶𝐶𝐴)   ,   𝑥 = 𝐴   ▶   𝑦(𝑦 ∈ {𝐴, 𝐵, 𝐶} ∧ 𝑦𝑥)   )
1413in2 39332 . 2 (   (𝐴𝐵𝐵𝐶𝐶𝐴)   ▶   (𝑥 = 𝐴 → ∃𝑦(𝑦 ∈ {𝐴, 𝐵, 𝐶} ∧ 𝑦𝑥))   )
1514in1 39289 1 ((𝐴𝐵𝐵𝐶𝐶𝐴) → (𝑥 = 𝐴 → ∃𝑦(𝑦 ∈ {𝐴, 𝐵, 𝐶} ∧ 𝑦𝑥)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 383  w3a 1072   = wceq 1632  wex 1853  wcel 2139  {ctp 4325
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1073  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-v 3342  df-un 3720  df-sn 4322  df-pr 4324  df-tp 4326  df-vd1 39288  df-vd2 39296
This theorem is referenced by:  en3lplem2VD  39578
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