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Theorem tpid3gVD 44840
Description: Virtual deduction proof of tpid3g 4777. (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 44611 . . . . . . 7 (   𝐴𝐵   ,   𝑥 = 𝐴   ▶   𝑥 = 𝐴   )
2 3mix3 1331 . . . . . . . . . 10 (𝑥 = 𝐴 → (𝑥 = 𝐶𝑥 = 𝐷𝑥 = 𝐴))
31, 2e2 44629 . . . . . . . . 9 (   𝐴𝐵   ,   𝑥 = 𝐴   ▶   (𝑥 = 𝐶𝑥 = 𝐷𝑥 = 𝐴)   )
4 abid 2716 . . . . . . . . 9 (𝑥 ∈ {𝑥 ∣ (𝑥 = 𝐶𝑥 = 𝐷𝑥 = 𝐴)} ↔ (𝑥 = 𝐶𝑥 = 𝐷𝑥 = 𝐴))
53, 4e2bir 44631 . . . . . . . 8 (   𝐴𝐵   ,   𝑥 = 𝐴   ▶   𝑥 ∈ {𝑥 ∣ (𝑥 = 𝐶𝑥 = 𝐷𝑥 = 𝐴)}   )
6 dftp2 4696 . . . . . . . . 9 {𝐶, 𝐷, 𝐴} = {𝑥 ∣ (𝑥 = 𝐶𝑥 = 𝐷𝑥 = 𝐴)}
76eleq2i 2831 . . . . . . . 8 (𝑥 ∈ {𝐶, 𝐷, 𝐴} ↔ 𝑥 ∈ {𝑥 ∣ (𝑥 = 𝐶𝑥 = 𝐷𝑥 = 𝐴)})
85, 7e2bir 44631 . . . . . . 7 (   𝐴𝐵   ,   𝑥 = 𝐴   ▶   𝑥 ∈ {𝐶, 𝐷, 𝐴}   )
9 eleq1 2827 . . . . . . . 8 (𝑥 = 𝐴 → (𝑥 ∈ {𝐶, 𝐷, 𝐴} ↔ 𝐴 ∈ {𝐶, 𝐷, 𝐴}))
109biimpd 229 . . . . . . 7 (𝑥 = 𝐴 → (𝑥 ∈ {𝐶, 𝐷, 𝐴} → 𝐴 ∈ {𝐶, 𝐷, 𝐴}))
111, 8, 10e22 44669 . . . . . 6 (   𝐴𝐵   ,   𝑥 = 𝐴   ▶   𝐴 ∈ {𝐶, 𝐷, 𝐴}   )
1211in2 44603 . . . . 5 (   𝐴𝐵   ▶   (𝑥 = 𝐴𝐴 ∈ {𝐶, 𝐷, 𝐴})   )
1312gen11 44614 . . . 4 (   𝐴𝐵   ▶   𝑥(𝑥 = 𝐴𝐴 ∈ {𝐶, 𝐷, 𝐴})   )
14 19.23v 1940 . . . 4 (∀𝑥(𝑥 = 𝐴𝐴 ∈ {𝐶, 𝐷, 𝐴}) ↔ (∃𝑥 𝑥 = 𝐴𝐴 ∈ {𝐶, 𝐷, 𝐴}))
1513, 14e1bi 44627 . . 3 (   𝐴𝐵   ▶   (∃𝑥 𝑥 = 𝐴𝐴 ∈ {𝐶, 𝐷, 𝐴})   )
16 idn1 44572 . . . 4 (   𝐴𝐵   ▶   𝐴𝐵   )
17 elisset 2821 . . . 4 (𝐴𝐵 → ∃𝑥 𝑥 = 𝐴)
1816, 17e1a 44625 . . 3 (   𝐴𝐵   ▶   𝑥 𝑥 = 𝐴   )
19 id 22 . . 3 ((∃𝑥 𝑥 = 𝐴𝐴 ∈ {𝐶, 𝐷, 𝐴}) → (∃𝑥 𝑥 = 𝐴𝐴 ∈ {𝐶, 𝐷, 𝐴}))
2015, 18, 19e11 44686 . 2 (   𝐴𝐵   ▶   𝐴 ∈ {𝐶, 𝐷, 𝐴}   )
2120in1 44569 1 (𝐴𝐵𝐴 ∈ {𝐶, 𝐷, 𝐴})
Colors of variables: wff setvar class
Syntax hints:  wi 4  w3o 1085  wal 1535   = wceq 1537  wex 1776  wcel 2106  {cab 2712  {ctp 4635
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-12 2175  ax-ext 2706
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-tru 1540  df-ex 1777  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-v 3480  df-un 3968  df-sn 4632  df-pr 4634  df-tp 4636  df-vd1 44568  df-vd2 44576
This theorem is referenced by: (None)
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