Theorem pm13.18 2647

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 2418	. . . 4 |- ( A = B -> ( A = C <-> B = C ) )
2	1	biimprd 215	. . 3 |- ( A = B -> ( B = C -> A = C ) )
3	2	necon3d 2613	. 2 |- ( A = B -> ( A =/= C -> B =/= C ) )
4	3	imp 419	1 |- ( ( A = B /\ A =/= C ) -> B =/= C )

Syntax hints:   -> wi 4   /\ wa 359   = wceq 1649   =/= wne 2575

This theorem is referenced by:  pm13.181  2648  4atexlemex4  30567

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-11 1757  ax-ext 2393

This theorem depends on definitions:  df-bi 178  df-an 361  df-ex 1548  df-cleq 2405  df-ne 2577