New Foundations Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  NFE Home  >  Th. List  >  ax12olem7 GIF version

Theorem ax12olem7 1933
 Description: Lemma for ax12o 1934. Derivation of ax12o 1934 from the hypotheses, without using ax12o 1934. (Contributed by NM, 24-Dec-2015.)
Hypotheses
Ref Expression
ax12olem7.1 x = z → (¬ x ¬ z = wx z = w))
ax12olem7.2 x = y → (¬ x ¬ y = wx y = w))
Assertion
Ref Expression
ax12olem7 x x = y → (¬ x x = z → (y = zx y = z)))
Distinct variable groups:   x,w   y,w   z,w

Proof of Theorem ax12olem7
StepHypRef Expression
1 ax12olem7.1 . . 3 x = z → (¬ x ¬ z = wx z = w))
21ax12olem5 1931 . 2 x x = z → (z = wx z = w))
3 ax12olem7.2 . . 3 x = y → (¬ x ¬ y = wx y = w))
43ax12olem5 1931 . 2 x x = y → (y = wx y = w))
52, 4ax12olem6 1932 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:  ax12o  1934
 Copyright terms: Public domain W3C validator