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Theorem tpid3g 3510
Description: Closed theorem form of tpid3 3511. (Contributed by Alan Sare, 24-Oct-2011.)
Assertion
Ref Expression
tpid3g (𝐴𝐵𝐴 ∈ {𝐶, 𝐷, 𝐴})

Proof of Theorem tpid3g
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 elisset 2585 . 2 (𝐴𝐵 → ∃𝑥 𝑥 = 𝐴)
2 3mix3 1086 . . . . . . 7 (𝑥 = 𝐴 → (𝑥 = 𝐶𝑥 = 𝐷𝑥 = 𝐴))
32a1i 9 . . . . . 6 (𝐴𝐵 → (𝑥 = 𝐴 → (𝑥 = 𝐶𝑥 = 𝐷𝑥 = 𝐴)))
4 abid 2044 . . . . . 6 (𝑥 ∈ {𝑥 ∣ (𝑥 = 𝐶𝑥 = 𝐷𝑥 = 𝐴)} ↔ (𝑥 = 𝐶𝑥 = 𝐷𝑥 = 𝐴))
53, 4syl6ibr 155 . . . . 5 (𝐴𝐵 → (𝑥 = 𝐴𝑥 ∈ {𝑥 ∣ (𝑥 = 𝐶𝑥 = 𝐷𝑥 = 𝐴)}))
6 dftp2 3446 . . . . . 6 {𝐶, 𝐷, 𝐴} = {𝑥 ∣ (𝑥 = 𝐶𝑥 = 𝐷𝑥 = 𝐴)}
76eleq2i 2120 . . . . 5 (𝑥 ∈ {𝐶, 𝐷, 𝐴} ↔ 𝑥 ∈ {𝑥 ∣ (𝑥 = 𝐶𝑥 = 𝐷𝑥 = 𝐴)})
85, 7syl6ibr 155 . . . 4 (𝐴𝐵 → (𝑥 = 𝐴𝑥 ∈ {𝐶, 𝐷, 𝐴}))
9 eleq1 2116 . . . 4 (𝑥 = 𝐴 → (𝑥 ∈ {𝐶, 𝐷, 𝐴} ↔ 𝐴 ∈ {𝐶, 𝐷, 𝐴}))
108, 9mpbidi 144 . . 3 (𝐴𝐵 → (𝑥 = 𝐴𝐴 ∈ {𝐶, 𝐷, 𝐴}))
1110exlimdv 1716 . 2 (𝐴𝐵 → (∃𝑥 𝑥 = 𝐴𝐴 ∈ {𝐶, 𝐷, 𝐴}))
121, 11mpd 13 1 (𝐴𝐵𝐴 ∈ {𝐶, 𝐷, 𝐴})
Colors of variables: wff set class
Syntax hints:  wi 4  w3o 895   = wceq 1259  wex 1397  wcel 1409  {cab 2042  {ctp 3404
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038
This theorem depends on definitions:  df-bi 114  df-3or 897  df-tru 1262  df-nf 1366  df-sb 1662  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-v 2576  df-un 2949  df-sn 3408  df-pr 3409  df-tp 3410
This theorem is referenced by: (None)
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