 New Foundations Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  NFE Home  >  Th. List  >  tpid3g GIF version

Theorem tpid3g 3831
 Description: Closed theorem form of tpid3 3832. This proof was automatically generated from the virtual deduction proof tpid3gVD in set.mm using a translation program. (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 2869 . 2 (A Bx x = A)
2 3mix3 1126 . . . . . . 7 (x = A → (x = C x = D x = A))
32a1i 10 . . . . . 6 (A B → (x = A → (x = C x = D x = A)))
4 abid 2341 . . . . . 6 (x {x (x = C x = D x = A)} ↔ (x = C x = D x = A))
53, 4syl6ibr 218 . . . . 5 (A B → (x = Ax {x (x = C x = D x = A)}))
6 dftp2 3772 . . . . . 6 {C, D, A} = {x (x = C x = D x = A)}
76eleq2i 2417 . . . . 5 (x {C, D, A} ↔ x {x (x = C x = D x = A)})
85, 7syl6ibr 218 . . . 4 (A B → (x = Ax {C, D, A}))
9 eleq1 2413 . . . 4 (x = A → (x {C, D, A} ↔ A {C, D, A}))
108, 9mpbidi 207 . . 3 (A B → (x = AA {C, D, A}))
1110exlimdv 1636 . 2 (A B → (x x = AA {C, D, A}))
121, 11mpd 14 1 (A BA {C, D, A})
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∨ w3o 933  ∃wex 1541   = wceq 1642   ∈ wcel 1710  {cab 2339  {ctp 3739 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334 This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2478  df-v 2861  df-nin 3211  df-compl 3212  df-un 3214  df-sn 3741  df-pr 3742  df-tp 3743 This theorem is referenced by: (None)
 Copyright terms: Public domain W3C validator