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Theorem ax12olem6 1932
 Description: Lemma for ax12o 1934. Derivation of ax12o 1934 from the hypotheses, without using ax12o 1934. (Contributed by Andrew Salmon, 21-Jul-2011.) (Revised by NM, 24-Dec-2015.)
Hypotheses
Ref Expression
ax12olem6.1 x x = z → (z = wx z = w))
ax12olem6.2 x x = y → (y = wx y = w))
Assertion
Ref Expression
ax12olem6 x x = y → (¬ x x = z → (y = zx y = z)))
Distinct variable groups:   x,w   y,w   z,w

Proof of Theorem ax12olem6
StepHypRef Expression
1 hbn1 1730 . . . . 5 x x = zx ¬ x x = z)
2 ax12olem6.1 . . . . 5 x x = z → (z = wx z = w))
31, 2hbim1 1810 . . . 4 ((¬ x x = zz = w) → xx x = zz = w))
4 ax-17 1616 . . . 4 ((¬ x x = zy = z) → wx x = zy = z))
5 equcom 1680 . . . . . 6 (z = ww = z)
6 equequ1 1684 . . . . . 6 (w = y → (w = zy = z))
75, 6syl5bb 248 . . . . 5 (w = y → (z = wy = z))
87imbi2d 307 . . . 4 (w = y → ((¬ x x = zz = w) ↔ (¬ x x = zy = z)))
9 ax12olem6.2 . . . 4 x x = y → (y = wx y = w))
103, 4, 8, 9dvelimhw 1849 . . 3 x x = y → ((¬ x x = zy = z) → xx x = zy = z)))
11119.21h 1797 . . 3 (xx x = zy = z) ↔ (¬ x x = zx y = z))
1210, 11syl6ib 217 . 2 x x = y → ((¬ x x = zy = z) → (¬ x x = zx y = z)))
1312pm2.86d 93 1 x x = y → (¬ x x = z → (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-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 This theorem depends on definitions:  df-bi 177  df-ex 1542  df-nf 1545 This theorem is referenced by:  ax12olem7  1933
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