Theorem pm13.18 2589

Description: Theorem *13.18 in [WhiteheadRussell] p. 178. (Contributed by Andrew Salmon, 3-Jun-2011.)

Assertion

Ref	Expression
pm13.18	⊢ ((A = B ∧ A ≠ C) → B ≠ C)

Proof of Theorem pm13.18

Step	Hyp	Ref	Expression
1		eqeq1 2359	. . . 4 ⊢ (A = B → (A = C ↔ B = C))
2	1	biimprd 214	. . 3 ⊢ (A = B → (B = C → A = C))
3	2	necon3d 2555	. 2 ⊢ (A = B → (A ≠ C → B ≠ C))
4	3	imp 418	1 ⊢ ((A = B ∧ A ≠ C) → B ≠ C)

This proof depends on syntax axioms:   → wi 4   ∧ wa 358   = wceq 1642   ≠ wne 2517

This proof depends on 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  ax-ext 2334

This proof depends on definitions:  df-bi 177  df-an 360  df-ex 1542  df-cleq 2346  df-ne 2519

This theorem is used by:  pm13.181  2590