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)

Syntax hints:   -> wi 4   /\ wa 358   = wceq 1642   =/= wne 2517

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-11 1746  ax-ext 2334

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

This theorem is referenced by:  pm13.181  2590