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Theorem tpid3gVD 44179
Description: Virtual deduction proof of tpid3g 4771. (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 43950 . . . . . . 7 (   𝐴𝐵   ,   𝑥 = 𝐴   ▶   𝑥 = 𝐴   )
2 3mix3 1329 . . . . . . . . . 10 (𝑥 = 𝐴 → (𝑥 = 𝐶𝑥 = 𝐷𝑥 = 𝐴))
31, 2e2 43968 . . . . . . . . 9 (   𝐴𝐵   ,   𝑥 = 𝐴   ▶   (𝑥 = 𝐶𝑥 = 𝐷𝑥 = 𝐴)   )
4 abid 2707 . . . . . . . . 9 (𝑥 ∈ {𝑥 ∣ (𝑥 = 𝐶𝑥 = 𝐷𝑥 = 𝐴)} ↔ (𝑥 = 𝐶𝑥 = 𝐷𝑥 = 𝐴))
53, 4e2bir 43970 . . . . . . . 8 (   𝐴𝐵   ,   𝑥 = 𝐴   ▶   𝑥 ∈ {𝑥 ∣ (𝑥 = 𝐶𝑥 = 𝐷𝑥 = 𝐴)}   )
6 dftp2 4688 . . . . . . . . 9 {𝐶, 𝐷, 𝐴} = {𝑥 ∣ (𝑥 = 𝐶𝑥 = 𝐷𝑥 = 𝐴)}
76eleq2i 2819 . . . . . . . 8 (𝑥 ∈ {𝐶, 𝐷, 𝐴} ↔ 𝑥 ∈ {𝑥 ∣ (𝑥 = 𝐶𝑥 = 𝐷𝑥 = 𝐴)})
85, 7e2bir 43970 . . . . . . 7 (   𝐴𝐵   ,   𝑥 = 𝐴   ▶   𝑥 ∈ {𝐶, 𝐷, 𝐴}   )
9 eleq1 2815 . . . . . . . 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 44008 . . . . . 6 (   𝐴𝐵   ,   𝑥 = 𝐴   ▶   𝐴 ∈ {𝐶, 𝐷, 𝐴}   )
1211in2 43942 . . . . 5 (   𝐴𝐵   ▶   (𝑥 = 𝐴𝐴 ∈ {𝐶, 𝐷, 𝐴})   )
1312gen11 43953 . . . 4 (   𝐴𝐵   ▶   𝑥(𝑥 = 𝐴𝐴 ∈ {𝐶, 𝐷, 𝐴})   )
14 19.23v 1937 . . . 4 (∀𝑥(𝑥 = 𝐴𝐴 ∈ {𝐶, 𝐷, 𝐴}) ↔ (∃𝑥 𝑥 = 𝐴𝐴 ∈ {𝐶, 𝐷, 𝐴}))
1513, 14e1bi 43966 . . 3 (   𝐴𝐵   ▶   (∃𝑥 𝑥 = 𝐴𝐴 ∈ {𝐶, 𝐷, 𝐴})   )
16 idn1 43911 . . . 4 (   𝐴𝐵   ▶   𝐴𝐵   )
17 elisset 2809 . . . 4 (𝐴𝐵 → ∃𝑥 𝑥 = 𝐴)
1816, 17e1a 43964 . . 3 (   𝐴𝐵   ▶   𝑥 𝑥 = 𝐴   )
19 id 22 . . 3 ((∃𝑥 𝑥 = 𝐴𝐴 ∈ {𝐶, 𝐷, 𝐴}) → (∃𝑥 𝑥 = 𝐴𝐴 ∈ {𝐶, 𝐷, 𝐴}))
2015, 18, 19e11 44025 . 2 (   𝐴𝐵   ▶   𝐴 ∈ {𝐶, 𝐷, 𝐴}   )
2120in1 43908 1 (𝐴𝐵𝐴 ∈ {𝐶, 𝐷, 𝐴})
Colors of variables: wff setvar class
Syntax hints:  wi 4  w3o 1083  wal 1531   = wceq 1533  wex 1773  wcel 2098  {cab 2703  {ctp 4627
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-12 2163  ax-ext 2697
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3or 1085  df-tru 1536  df-ex 1774  df-sb 2060  df-clab 2704  df-cleq 2718  df-clel 2804  df-v 3470  df-un 3948  df-sn 4624  df-pr 4626  df-tp 4628  df-vd1 43907  df-vd2 43915
This theorem is referenced by: (None)
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