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Theorem tpid3gVD 38099
Description: Virtual deduction proof of tpid3g 4248. (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 37859 . . . . . . 7 (   𝐴𝐵   ,   𝑥 = 𝐴   ▶   𝑥 = 𝐴   )
2 3mix3 1225 . . . . . . . . . 10 (𝑥 = 𝐴 → (𝑥 = 𝐶𝑥 = 𝐷𝑥 = 𝐴))
31, 2e2 37877 . . . . . . . . 9 (   𝐴𝐵   ,   𝑥 = 𝐴   ▶   (𝑥 = 𝐶𝑥 = 𝐷𝑥 = 𝐴)   )
4 abid 2598 . . . . . . . . 9 (𝑥 ∈ {𝑥 ∣ (𝑥 = 𝐶𝑥 = 𝐷𝑥 = 𝐴)} ↔ (𝑥 = 𝐶𝑥 = 𝐷𝑥 = 𝐴))
53, 4e2bir 37879 . . . . . . . 8 (   𝐴𝐵   ,   𝑥 = 𝐴   ▶   𝑥 ∈ {𝑥 ∣ (𝑥 = 𝐶𝑥 = 𝐷𝑥 = 𝐴)}   )
6 dftp2 4178 . . . . . . . . 9 {𝐶, 𝐷, 𝐴} = {𝑥 ∣ (𝑥 = 𝐶𝑥 = 𝐷𝑥 = 𝐴)}
76eleq2i 2680 . . . . . . . 8 (𝑥 ∈ {𝐶, 𝐷, 𝐴} ↔ 𝑥 ∈ {𝑥 ∣ (𝑥 = 𝐶𝑥 = 𝐷𝑥 = 𝐴)})
85, 7e2bir 37879 . . . . . . 7 (   𝐴𝐵   ,   𝑥 = 𝐴   ▶   𝑥 ∈ {𝐶, 𝐷, 𝐴}   )
9 eleq1 2676 . . . . . . . 8 (𝑥 = 𝐴 → (𝑥 ∈ {𝐶, 𝐷, 𝐴} ↔ 𝐴 ∈ {𝐶, 𝐷, 𝐴}))
109biimpd 218 . . . . . . 7 (𝑥 = 𝐴 → (𝑥 ∈ {𝐶, 𝐷, 𝐴} → 𝐴 ∈ {𝐶, 𝐷, 𝐴}))
111, 8, 10e22 37917 . . . . . 6 (   𝐴𝐵   ,   𝑥 = 𝐴   ▶   𝐴 ∈ {𝐶, 𝐷, 𝐴}   )
1211in2 37851 . . . . 5 (   𝐴𝐵   ▶   (𝑥 = 𝐴𝐴 ∈ {𝐶, 𝐷, 𝐴})   )
1312gen11 37862 . . . 4 (   𝐴𝐵   ▶   𝑥(𝑥 = 𝐴𝐴 ∈ {𝐶, 𝐷, 𝐴})   )
14 19.23v 1889 . . . 4 (∀𝑥(𝑥 = 𝐴𝐴 ∈ {𝐶, 𝐷, 𝐴}) ↔ (∃𝑥 𝑥 = 𝐴𝐴 ∈ {𝐶, 𝐷, 𝐴}))
1513, 14e1bi 37875 . . 3 (   𝐴𝐵   ▶   (∃𝑥 𝑥 = 𝐴𝐴 ∈ {𝐶, 𝐷, 𝐴})   )
16 idn1 37811 . . . 4 (   𝐴𝐵   ▶   𝐴𝐵   )
17 elisset 3188 . . . 4 (𝐴𝐵 → ∃𝑥 𝑥 = 𝐴)
1816, 17e1a 37873 . . 3 (   𝐴𝐵   ▶   𝑥 𝑥 = 𝐴   )
19 id 22 . . 3 ((∃𝑥 𝑥 = 𝐴𝐴 ∈ {𝐶, 𝐷, 𝐴}) → (∃𝑥 𝑥 = 𝐴𝐴 ∈ {𝐶, 𝐷, 𝐴}))
2015, 18, 19e11 37934 . 2 (   𝐴𝐵   ▶   𝐴 ∈ {𝐶, 𝐷, 𝐴}   )
2120in1 37808 1 (𝐴𝐵𝐴 ∈ {𝐶, 𝐷, 𝐴})
Colors of variables: wff setvar class
Syntax hints:  wi 4  w3o 1030  wal 1473   = wceq 1475  wex 1695  wcel 1977  {cab 2596  {ctp 4129
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590
This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-v 3175  df-un 3545  df-sn 4126  df-pr 4128  df-tp 4130  df-vd1 37807  df-vd2 37815
This theorem is referenced by: (None)
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