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Theorem ax12olem3 1929
 Description: Lemma for ax12o 1934. Show the equivalence of an intermediate equivalent to ax12o 1934 with the conjunction of ax-12 1925 and a variant with negated equalities. (Contributed by NM, 24-Dec-2015.)
Assertion
Ref Expression
ax12olem3 ((¬ x = y → (¬ x ¬ y = zx y = z)) ↔ ((¬ x = y → (y = zx y = z)) x = y → (¬ y = zx ¬ y = z))))

Proof of Theorem ax12olem3
StepHypRef Expression
1 sp 1747 . . . . . 6 (x ¬ y = z → ¬ y = z)
21con2i 112 . . . . 5 (y = z → ¬ x ¬ y = z)
32imim1i 54 . . . 4 ((¬ x ¬ y = zx y = z) → (y = zx y = z))
43imim2i 13 . . 3 ((¬ x = y → (¬ x ¬ y = zx y = z)) → (¬ x = y → (y = zx y = z)))
5 sp 1747 . . . . . 6 (x y = zy = z)
65imim2i 13 . . . . 5 ((¬ x ¬ y = zx y = z) → (¬ x ¬ y = zy = z))
76con1d 116 . . . 4 ((¬ x ¬ y = zx y = z) → (¬ y = zx ¬ y = z))
87imim2i 13 . . 3 ((¬ x = y → (¬ x ¬ y = zx y = z)) → (¬ x = y → (¬ y = zx ¬ y = z)))
94, 8jca 518 . 2 ((¬ x = y → (¬ x ¬ y = zx y = z)) → ((¬ x = y → (y = zx y = z)) x = y → (¬ y = zx ¬ y = z))))
10 con1 120 . . . . . 6 ((¬ y = zx ¬ y = z) → (¬ x ¬ y = zy = z))
1110imim1d 69 . . . . 5 ((¬ y = zx ¬ y = z) → ((y = zx y = z) → (¬ x ¬ y = zx y = z)))
1211com12 27 . . . 4 ((y = zx y = z) → ((¬ y = zx ¬ y = z) → (¬ x ¬ y = zx y = z)))
1312imim3i 55 . . 3 ((¬ x = y → (y = zx y = z)) → ((¬ x = y → (¬ y = zx ¬ y = z)) → (¬ x = y → (¬ x ¬ y = zx y = z))))
1413imp 418 . 2 (((¬ x = y → (y = zx y = z)) x = y → (¬ y = zx ¬ y = z))) → (¬ x = y → (¬ x ¬ y = zx y = z)))
159, 14impbii 180 1 ((¬ x = y → (¬ x ¬ y = zx y = z)) ↔ ((¬ x = y → (y = zx y = z)) x = y → (¬ y = zx ¬ y = z))))
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 176   ∧ wa 358  ∀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-11 1746 This theorem depends on definitions:  df-bi 177  df-an 360  df-ex 1542 This theorem is referenced by:  ax12olem4  1930
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