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Theorem ax12from12o 2156
 Description: Derive ax-12 1925 from ax-12o 2142 and other older axioms. This proof uses newer axioms ax-5 1557 and ax-9 1654, but since these are proved from the older axioms above, this is acceptable and lets us avoid having to reprove several earlier theorems to use ax-5o 2136 and ax-9o 2138. (Contributed by NM, 21-Dec-2015.) (Proof modification is discouraged.) (New usage is discouraged.)
Assertion
Ref Expression
ax12from12o x = y → (y = zx y = z))

Proof of Theorem ax12from12o
StepHypRef Expression
1 ax-4 2135 . . . . . 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 ax-4 2135 . . . . 5 (x x = zx = z)
97, 8nsyl 113 . . . 4 ((¬ x = y y = z) → ¬ x x = z)
10 ax-12o 2142 . . . 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-4 2135  ax-12o 2142 This theorem depends on definitions:  df-bi 177  df-an 360  df-ex 1542 This theorem is referenced by: (None)
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