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Theorem tpid3gOLD 4249
 Description: Obsolete proof of tpid3g 4248 as of 30-Apr-2021. Closed theorem form of tpid3 4250. This proof was automatically generated from the virtual deduction proof tpid3gVD 38099 using a translation program. (Contributed by Alan Sare, 24-Oct-2011.) (New usage is discouraged.) (Proof modification is discouraged.)
Assertion
Ref Expression
tpid3gOLD (𝐴𝐵𝐴 ∈ {𝐶, 𝐷, 𝐴})

Proof of Theorem tpid3gOLD
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 elisset 3188 . 2 (𝐴𝐵 → ∃𝑥 𝑥 = 𝐴)
2 3mix3 1225 . . . . . . 7 (𝑥 = 𝐴 → (𝑥 = 𝐶𝑥 = 𝐷𝑥 = 𝐴))
32a1i 11 . . . . . 6 (𝐴𝐵 → (𝑥 = 𝐴 → (𝑥 = 𝐶𝑥 = 𝐷𝑥 = 𝐴)))
4 abid 2598 . . . . . 6 (𝑥 ∈ {𝑥 ∣ (𝑥 = 𝐶𝑥 = 𝐷𝑥 = 𝐴)} ↔ (𝑥 = 𝐶𝑥 = 𝐷𝑥 = 𝐴))
53, 4syl6ibr 241 . . . . 5 (𝐴𝐵 → (𝑥 = 𝐴𝑥 ∈ {𝑥 ∣ (𝑥 = 𝐶𝑥 = 𝐷𝑥 = 𝐴)}))
6 dftp2 4178 . . . . . 6 {𝐶, 𝐷, 𝐴} = {𝑥 ∣ (𝑥 = 𝐶𝑥 = 𝐷𝑥 = 𝐴)}
76eleq2i 2680 . . . . 5 (𝑥 ∈ {𝐶, 𝐷, 𝐴} ↔ 𝑥 ∈ {𝑥 ∣ (𝑥 = 𝐶𝑥 = 𝐷𝑥 = 𝐴)})
85, 7syl6ibr 241 . . . 4 (𝐴𝐵 → (𝑥 = 𝐴𝑥 ∈ {𝐶, 𝐷, 𝐴}))
9 eleq1 2676 . . . 4 (𝑥 = 𝐴 → (𝑥 ∈ {𝐶, 𝐷, 𝐴} ↔ 𝐴 ∈ {𝐶, 𝐷, 𝐴}))
108, 9mpbidi 230 . . 3 (𝐴𝐵 → (𝑥 = 𝐴𝐴 ∈ {𝐶, 𝐷, 𝐴}))
1110exlimdv 1848 . 2 (𝐴𝐵 → (∃𝑥 𝑥 = 𝐴𝐴 ∈ {𝐶, 𝐷, 𝐴}))
121, 11mpd 15 1 (𝐴𝐵𝐴 ∈ {𝐶, 𝐷, 𝐴})
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∨ w3o 1030   = 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 This theorem is referenced by: (None)
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