Theorem pm13.18 2389

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 2146	. . . 4 |- ( A = B -> ( A = C <-> B = C ) )
2	1	biimprd 157	. . 3 |- ( A = B -> ( B = C -> A = C ) )
3	2	necon3d 2352	. 2 |- ( A = B -> ( A =/= C -> B =/= C ) )
4	3	imp 123	1 |- ( ( A = B /\ A =/= C ) -> B =/= C )

Syntax hints:   -> wi 4   /\ wa 103   = wceq 1331   =/= wne 2308

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 603  ax-in2 604  ax-5 1423  ax-gen 1425  ax-4 1487  ax-17 1506  ax-ext 2121

This theorem depends on definitions:  df-bi 116  df-cleq 2132  df-ne 2309

This theorem is referenced by:  pm13.181  2390