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Theorem tpid3g 3474
Description: Closed theorem form of tpid3 3475. (Contributed by Alan Sare, 24-Oct-2011.)
Assertion
Ref Expression
tpid3g (A BA {𝐶, 𝐷, A})

Proof of Theorem tpid3g
Dummy variable x is distinct from all other variables.
StepHypRef Expression
1 elisset 2562 . 2 (A Bx x = A)
2 3mix3 1074 . . . . . . 7 (x = A → (x = 𝐶 x = 𝐷 x = A))
32a1i 9 . . . . . 6 (A B → (x = A → (x = 𝐶 x = 𝐷 x = A)))
4 abid 2025 . . . . . 6 (x {x ∣ (x = 𝐶 x = 𝐷 x = A)} ↔ (x = 𝐶 x = 𝐷 x = A))
53, 4syl6ibr 151 . . . . 5 (A B → (x = Ax {x ∣ (x = 𝐶 x = 𝐷 x = A)}))
6 dftp2 3410 . . . . . 6 {𝐶, 𝐷, A} = {x ∣ (x = 𝐶 x = 𝐷 x = A)}
76eleq2i 2101 . . . . 5 (x {𝐶, 𝐷, A} ↔ x {x ∣ (x = 𝐶 x = 𝐷 x = A)})
85, 7syl6ibr 151 . . . 4 (A B → (x = Ax {𝐶, 𝐷, A}))
9 eleq1 2097 . . . 4 (x = A → (x {𝐶, 𝐷, A} ↔ A {𝐶, 𝐷, A}))
108, 9mpbidi 140 . . 3 (A B → (x = AA {𝐶, 𝐷, A}))
1110exlimdv 1697 . 2 (A B → (x x = AA {𝐶, 𝐷, A}))
121, 11mpd 13 1 (A BA {𝐶, 𝐷, A})
Colors of variables: wff set class
Syntax hints:  wi 4   w3o 883   = wceq 1242  wex 1378   wcel 1390  {cab 2023  {ctp 3369
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bnd 1396  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019
This theorem depends on definitions:  df-bi 110  df-3or 885  df-tru 1245  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-v 2553  df-un 2916  df-sn 3373  df-pr 3374  df-tp 3375
This theorem is referenced by: (None)
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