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Theorem ax12olem5 1931
 Description: Lemma for ax12o 1934. See ax12olem6 1932 for derivation of ax12o 1934 from the conclusion. (Contributed by NM, 24-Dec-2015.)
Hypothesis
Ref Expression
ax12olem5.1 x = y → (¬ x ¬ y = zx y = z))
Assertion
Ref Expression
ax12olem5 x x = y → (y = zx y = z))

Proof of Theorem ax12olem5
StepHypRef Expression
1 exnal 1574 . 2 (x ¬ x = y ↔ ¬ x x = y)
2 19.8a 1756 . . 3 (y = zx y = z)
3 hbe1 1731 . . . . 5 (x y = zxx y = z)
4 hba1 1786 . . . . 5 (x y = zxx y = z)
53, 4hbim 1817 . . . 4 ((x y = zx y = z) → x(x y = zx y = z))
6 df-ex 1542 . . . . 5 (x y = z ↔ ¬ x ¬ y = z)
7 ax12olem5.1 . . . . 5 x = y → (¬ x ¬ y = zx y = z))
86, 7syl5bi 208 . . . 4 x = y → (x y = zx y = z))
95, 8exlimih 1804 . . 3 (x ¬ x = y → (x y = zx y = z))
102, 9syl5 28 . 2 (x ¬ x = y → (y = zx y = z))
111, 10sylbir 204 1 x x = y → (y = zx y = z))
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4  ∀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-11 1746 This theorem depends on definitions:  df-bi 177  df-ex 1542  df-nf 1545 This theorem is referenced by:  ax12olem7  1933
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