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Theorem ax12olem4 1930
 Description: Lemma for ax12o 1934. Construct an intermediate equivalent to ax-12 1925 from two instances of ax-12 1925. (Contributed by NM, 24-Dec-2015.)
Hypotheses
Ref Expression
ax12olem4.1 x = y → (y = zx y = z))
ax12olem4.2 x = y → (y = wx y = w))
Assertion
Ref Expression
ax12olem4 x = y → (¬ x ¬ y = zx y = z))
Distinct variable groups:   x,w,z   y,w,z

Proof of Theorem ax12olem4
StepHypRef Expression
1 ax12olem4.1 . 2 x = y → (y = zx y = z))
2 ax12olem4.2 . . 3 x = y → (y = wx y = w))
32ax12olem2 1928 . 2 x = y → (¬ y = zx ¬ y = z))
4 ax12olem3 1929 . 2 ((¬ x = y → (¬ x ¬ y = zx y = z)) ↔ ((¬ x = y → (y = zx y = z)) x = y → (¬ y = zx ¬ y = z))))
51, 3, 4mpbir2an 886 1 x = y → (¬ x ¬ y = zx y = z))
 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 This theorem depends on definitions:  df-bi 177  df-an 360  df-ex 1542  df-nf 1545 This theorem is referenced by:  ax12o  1934
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