New Foundations Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  NFE Home  >  Th. List  >  ax12olem2 GIF version

Theorem ax12olem2 1928
 Description: Lemma for ax12o 1934. Negate the equalities in ax-12 1925, shown as the hypothesis. (Contributed by NM, 24-Dec-2015.)
Hypothesis
Ref Expression
ax12olem2.1 x = y → (y = wx y = w))
Assertion
Ref Expression
ax12olem2 x = y → (¬ y = zx ¬ y = z))
Distinct variable groups:   x,w,z   y,w

Proof of Theorem ax12olem2
StepHypRef Expression
1 ax12olem2.1 . . . . . 6 x = y → (y = wx y = w))
21anim1d 547 . . . . 5 x = y → ((y = w ¬ z = w) → (x y = w ¬ z = w)))
3 ax-17 1616 . . . . . . 7 z = wx ¬ z = w)
43anim2i 552 . . . . . 6 ((x y = w ¬ z = w) → (x y = w x ¬ z = w))
5 19.26 1593 . . . . . 6 (x(y = w ¬ z = w) ↔ (x y = w x ¬ z = w))
64, 5sylibr 203 . . . . 5 ((x y = w ¬ z = w) → x(y = w ¬ z = w))
72, 6syl6 29 . . . 4 x = y → ((y = w ¬ z = w) → x(y = w ¬ z = w)))
87eximdv 1622 . . 3 x = y → (w(y = w ¬ z = w) → wx(y = w ¬ z = w)))
9 19.12 1847 . . 3 (wx(y = w ¬ z = w) → xw(y = w ¬ z = w))
108, 9syl6 29 . 2 x = y → (w(y = w ¬ z = w) → xw(y = w ¬ z = w)))
11 ax12olem1 1927 . 2 (w(y = w ¬ z = w) ↔ ¬ y = z)
1211albii 1566 . 2 (xw(y = w ¬ z = w) ↔ x ¬ y = z)
1310, 11, 123imtr3g 260 1 x = y → (¬ y = zx ¬ y = z))
 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 This theorem depends on definitions:  df-bi 177  df-an 360  df-ex 1542  df-nf 1545 This theorem is referenced by:  ax12olem4  1930
 Copyright terms: Public domain W3C validator