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Theorem ax12b 1689
 Description: Two equivalent ways of expressing ax-12 1925. See the comment for ax-12 1925. (Contributed by NM, 2-May-2017.) (Proof shortened by Wolf Lammen, 12-Aug-2017.)
Assertion
Ref Expression
ax12b ((¬ x = y → (y = zx y = z)) ↔ (¬ x = y → (¬ x = z → (y = zx y = z))))

Proof of Theorem ax12b
StepHypRef Expression
1 id 19 . . 3 ((¬ x = y → (y = zx y = z)) → (¬ x = y → (y = zx y = z)))
21a1dd 42 . 2 ((¬ x = y → (y = zx y = z)) → (¬ x = y → (¬ x = z → (y = zx y = z))))
3 equtrr 1683 . . . . . 6 (z = y → (x = zx = y))
43equcoms 1681 . . . . 5 (y = z → (x = zx = y))
54con3rr3 128 . . . 4 x = y → (y = z → ¬ x = z))
6 id 19 . . . . . 6 ((¬ x = y → (¬ x = z → (y = zx y = z))) → (¬ x = y → (¬ x = z → (y = zx y = z))))
76com4l 78 . . . . 5 x = y → (¬ x = z → (y = z → ((¬ x = y → (¬ x = z → (y = zx y = z))) → x y = z))))
87com23 72 . . . 4 x = y → (y = z → (¬ x = z → ((¬ x = y → (¬ x = z → (y = zx y = z))) → x y = z))))
95, 8mpdd 36 . . 3 x = y → (y = z → ((¬ x = y → (¬ x = z → (y = zx y = z))) → x y = z)))
109com3r 73 . 2 ((¬ x = y → (¬ x = z → (y = zx y = z))) → (¬ x = y → (y = zx y = z)))
112, 10impbii 180 1 ((¬ x = y → (y = zx y = z)) ↔ (¬ x = y → (¬ x = z → (y = zx y = z))))
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 176  ∀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 This theorem depends on definitions:  df-bi 177  df-ex 1542 This theorem is referenced by: (None)
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