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Theorem ax12o 1934
 Description: Derive set.mm's original ax-12o 2142 from the shorter ax-12 1925. (Contributed by NM, 29-Nov-2015.) (Revised by NM, 24-Dec-2015.)
Assertion
Ref Expression
ax12o z z = x → (¬ z z = y → (x = yz x = y)))

Proof of Theorem ax12o
Dummy variables w v are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 ax12v 1926 . . 3 z = y → (y = wz y = w))
2 ax12v 1926 . . 3 z = y → (y = vz y = v))
31, 2ax12olem4 1930 . 2 z = y → (¬ z ¬ y = wz y = w))
4 ax12v 1926 . . 3 z = x → (x = wz x = w))
5 ax12v 1926 . . 3 z = x → (x = vz x = v))
64, 5ax12olem4 1930 . 2 z = x → (¬ z ¬ x = wz x = w))
73, 6ax12olem7 1933 1 z z = x → (¬ z z = y → (x = yz x = y)))
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4  ∀wal 1540 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 This theorem depends on definitions:  df-bi 177  df-an 360  df-ex 1542  df-nf 1545 This theorem is referenced by:  ax12  1935  dvelimv  1939  hbae  1953  nfeqf  1958  dvelimh  1964  dvelimf  1997  dvelimALT  2133  ax11eq  2193  ax11indalem  2197  axi12  2333
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