Theorem pm13.18 2678

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 2444	. . . 4 |- ( A = B -> ( A = C <-> B = C ) )
2	1	biimprd 216	. . 3 |- ( A = B -> ( B = C -> A = C ) )
3	2	necon3d 2641	. 2 |- ( A = B -> ( A =/= C -> B =/= C ) )
4	3	imp 420	1 |- ( ( A = B /\ A =/= C ) -> B =/= C )

Syntax hints:   -> wi 4   /\ wa 360   = wceq 1653   =/= wne 2601

This theorem is referenced by:  pm13.181  2679  4atexlemex4  30944

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-11 1762  ax-ext 2419

This theorem depends on definitions:  df-bi 179  df-an 362  df-ex 1552  df-cleq 2431  df-ne 2603