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Theorem tpid3gVD 43535
Description: Virtual deduction proof of tpid3g 4774. (Contributed by Alan Sare, 24-Oct-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
Assertion
Ref Expression
tpid3gVD (𝐴𝐵𝐴 ∈ {𝐶, 𝐷, 𝐴})

Proof of Theorem tpid3gVD
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 idn2 43306 . . . . . . 7 (   𝐴𝐵   ,   𝑥 = 𝐴   ▶   𝑥 = 𝐴   )
2 3mix3 1333 . . . . . . . . . 10 (𝑥 = 𝐴 → (𝑥 = 𝐶𝑥 = 𝐷𝑥 = 𝐴))
31, 2e2 43324 . . . . . . . . 9 (   𝐴𝐵   ,   𝑥 = 𝐴   ▶   (𝑥 = 𝐶𝑥 = 𝐷𝑥 = 𝐴)   )
4 abid 2714 . . . . . . . . 9 (𝑥 ∈ {𝑥 ∣ (𝑥 = 𝐶𝑥 = 𝐷𝑥 = 𝐴)} ↔ (𝑥 = 𝐶𝑥 = 𝐷𝑥 = 𝐴))
53, 4e2bir 43326 . . . . . . . 8 (   𝐴𝐵   ,   𝑥 = 𝐴   ▶   𝑥 ∈ {𝑥 ∣ (𝑥 = 𝐶𝑥 = 𝐷𝑥 = 𝐴)}   )
6 dftp2 4691 . . . . . . . . 9 {𝐶, 𝐷, 𝐴} = {𝑥 ∣ (𝑥 = 𝐶𝑥 = 𝐷𝑥 = 𝐴)}
76eleq2i 2826 . . . . . . . 8 (𝑥 ∈ {𝐶, 𝐷, 𝐴} ↔ 𝑥 ∈ {𝑥 ∣ (𝑥 = 𝐶𝑥 = 𝐷𝑥 = 𝐴)})
85, 7e2bir 43326 . . . . . . 7 (   𝐴𝐵   ,   𝑥 = 𝐴   ▶   𝑥 ∈ {𝐶, 𝐷, 𝐴}   )
9 eleq1 2822 . . . . . . . 8 (𝑥 = 𝐴 → (𝑥 ∈ {𝐶, 𝐷, 𝐴} ↔ 𝐴 ∈ {𝐶, 𝐷, 𝐴}))
109biimpd 228 . . . . . . 7 (𝑥 = 𝐴 → (𝑥 ∈ {𝐶, 𝐷, 𝐴} → 𝐴 ∈ {𝐶, 𝐷, 𝐴}))
111, 8, 10e22 43364 . . . . . 6 (   𝐴𝐵   ,   𝑥 = 𝐴   ▶   𝐴 ∈ {𝐶, 𝐷, 𝐴}   )
1211in2 43298 . . . . 5 (   𝐴𝐵   ▶   (𝑥 = 𝐴𝐴 ∈ {𝐶, 𝐷, 𝐴})   )
1312gen11 43309 . . . 4 (   𝐴𝐵   ▶   𝑥(𝑥 = 𝐴𝐴 ∈ {𝐶, 𝐷, 𝐴})   )
14 19.23v 1946 . . . 4 (∀𝑥(𝑥 = 𝐴𝐴 ∈ {𝐶, 𝐷, 𝐴}) ↔ (∃𝑥 𝑥 = 𝐴𝐴 ∈ {𝐶, 𝐷, 𝐴}))
1513, 14e1bi 43322 . . 3 (   𝐴𝐵   ▶   (∃𝑥 𝑥 = 𝐴𝐴 ∈ {𝐶, 𝐷, 𝐴})   )
16 idn1 43267 . . . 4 (   𝐴𝐵   ▶   𝐴𝐵   )
17 elisset 2816 . . . 4 (𝐴𝐵 → ∃𝑥 𝑥 = 𝐴)
1816, 17e1a 43320 . . 3 (   𝐴𝐵   ▶   𝑥 𝑥 = 𝐴   )
19 id 22 . . 3 ((∃𝑥 𝑥 = 𝐴𝐴 ∈ {𝐶, 𝐷, 𝐴}) → (∃𝑥 𝑥 = 𝐴𝐴 ∈ {𝐶, 𝐷, 𝐴}))
2015, 18, 19e11 43381 . 2 (   𝐴𝐵   ▶   𝐴 ∈ {𝐶, 𝐷, 𝐴}   )
2120in1 43264 1 (𝐴𝐵𝐴 ∈ {𝐶, 𝐷, 𝐴})
Colors of variables: wff setvar class
Syntax hints:  wi 4  w3o 1087  wal 1540   = wceq 1542  wex 1782  wcel 2107  {cab 2710  {ctp 4630
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-12 2172  ax-ext 2704
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  df-tru 1545  df-ex 1783  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-v 3477  df-un 3951  df-sn 4627  df-pr 4629  df-tp 4631  df-vd1 43263  df-vd2 43271
This theorem is referenced by: (None)
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