Theorem pm13.181 2589

Description: Theorem *13.181 in [WhiteheadRussell] p. 178. (Contributed by Andrew Salmon, 3-Jun-2011.)

Assertion

Ref	Expression
pm13.181	⊢ ((A = B ∧ B ≠ C) → A ≠ C)

Proof of Theorem pm13.181

Step	Hyp	Ref	Expression
1		eqcom 2355	. 2 ⊢ (A = B ↔ B = A)
2		pm13.18 2588	. 2 ⊢ ((B = A ∧ B ≠ C) → A ≠ C)
3	1, 2	sylanb 458	1 ⊢ ((A = B ∧ B ≠ C) → A ≠ C)

Syntax hints:   → wi 4   ∧ wa 358   = wceq 1642   ≠ wne 2516

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-11 1746  ax-ext 2334

This theorem depends on definitions:  df-bi 177  df-an 360  df-ex 1542  df-cleq 2346  df-ne 2518

This theorem is referenced by: (None)