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Theorem tpid3g 3453
 Description: Closed theorem form of tpid3 3454. (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 2541 . 2 (A Bx x = A)
2 3mix3 1061 . . . . . . 7 (x = A → (x = 𝐶 x = 𝐷 x = A))
32a1i 9 . . . . . 6 (A B → (x = A → (x = 𝐶 x = 𝐷 x = A)))
4 abid 2006 . . . . . 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 3389 . . . . . 6 {𝐶, 𝐷, A} = {x ∣ (x = 𝐶 x = 𝐷 x = A)}
76eleq2i 2082 . . . . 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 2078 . . . 4 (x = A → (x {𝐶, 𝐷, A} ↔ A {𝐶, 𝐷, A}))
108, 9mpbidi 140 . . 3 (A B → (x = AA {𝐶, 𝐷, A}))
1110exlimdv 1678 . 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 870   = wceq 1226  ∃wex 1358   ∈ wcel 1370  {cab 2004  {ctp 3348 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 617  ax-5 1312  ax-7 1313  ax-gen 1314  ax-ie1 1359  ax-ie2 1360  ax-8 1372  ax-10 1373  ax-11 1374  ax-i12 1375  ax-bnd 1376  ax-4 1377  ax-17 1396  ax-i9 1400  ax-ial 1405  ax-i5r 1406  ax-ext 2000 This theorem depends on definitions:  df-bi 110  df-3or 872  df-tru 1229  df-nf 1326  df-sb 1624  df-clab 2005  df-cleq 2011  df-clel 2014  df-nfc 2145  df-v 2533  df-un 2895  df-sn 3352  df-pr 3353  df-tp 3354 This theorem is referenced by: (None)
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