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Theorem ax12 1935
 Description: Derive ax-12 1925 from ax12v 1926 via ax12o 1934. This shows that the weakening in ax12v 1926 is still sufficient for a complete system. (Contributed by NM, 21-Dec-2015.)
Assertion
Ref Expression
ax12 x = y → (y = zx y = z))

Proof of Theorem ax12
StepHypRef Expression
1 sp 1747 . . . . . 6 (x x = yx = y)
21con3i 127 . . . . 5 x = y → ¬ x x = y)
32adantr 451 . . . 4 ((¬ x = y y = z) → ¬ x x = y)
4 equtrr 1683 . . . . . . . 8 (z = y → (x = zx = y))
54equcoms 1681 . . . . . . 7 (y = z → (x = zx = y))
65con3rr3 128 . . . . . 6 x = y → (y = z → ¬ x = z))
76imp 418 . . . . 5 ((¬ x = y y = z) → ¬ x = z)
8 sp 1747 . . . . 5 (x x = zx = z)
97, 8nsyl 113 . . . 4 ((¬ x = y y = z) → ¬ x x = z)
10 ax12o 1934 . . . 4 x x = y → (¬ x x = z → (y = zx y = z)))
113, 9, 10sylc 56 . . 3 ((¬ x = y y = z) → (y = zx y = z))
1211ex 423 . 2 x = y → (y = z → (y = zx y = z)))
1312pm2.43d 44 1 x = y → (y = zx y = z))
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 358  ∀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  ax-12 1925 This theorem depends on definitions:  df-bi 177  df-an 360  df-ex 1542  df-nf 1545 This theorem is referenced by: (None)
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