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Theorem ax10 1944
 Description: Derive set.mm's original ax-10 2140 from others. (Contributed by NM, 25-Jul-2015.) (Revised by NM, 7-Nov-2015.)
Assertion
Ref Expression
ax10 (x x = yy y = x)

Proof of Theorem ax10
Dummy variable z is distinct from all other variables.
StepHypRef Expression
1 ax9v 1655 . 2 ¬ z ¬ z = x
2 df-ex 1542 . . 3 (z z = x ↔ ¬ z ¬ z = x)
3 dveeq2 1940 . . . . . . . 8 y y = x → (z = xy z = x))
43imp 418 . . . . . . 7 ((¬ y y = x z = x) → y z = x)
5 ax10lem6 1943 . . . . . . . 8 (x x = y → (y z = xx z = x))
6 equcomi 1679 . . . . . . . . 9 (z = xx = z)
76alimi 1559 . . . . . . . 8 (x z = xx x = z)
85, 7syl6 29 . . . . . . 7 (x x = y → (y z = xx x = z))
9 ax10lem5 1942 . . . . . . 7 (x x = zy y = x)
104, 8, 9syl56 30 . . . . . 6 (x x = y → ((¬ y y = x z = x) → y y = x))
1110exp3acom23 1372 . . . . 5 (x x = y → (z = x → (¬ y y = xy y = x)))
12 pm2.18 102 . . . . 5 ((¬ y y = xy y = x) → y y = x)
1311, 12syl6 29 . . . 4 (x x = y → (z = xy y = x))
1413exlimdv 1636 . . 3 (x x = y → (z z = xy y = x))
152, 14syl5bir 209 . 2 (x x = y → (¬ z ¬ z = xy y = x))
161, 15mpi 16 1 (x x = yy y = x)
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 358  ∀wal 1540  ∃wex 1541 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 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 This theorem depends on definitions:  df-bi 177  df-an 360  df-tru 1319  df-ex 1542  df-nf 1545 This theorem is referenced by:  aecom  1946  ax10o  1952  axi10  2331
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