Theorem pm13.18 2326

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 2087	. . . 4 |- ( A = B -> ( A = C <-> B = C ) )
2	1	biimprd 156	. . 3 |- ( A = B -> ( B = C -> A = C ) )
3	2	necon3d 2289	. 2 |- ( A = B -> ( A =/= C -> B =/= C ) )
4	3	imp 122	1 |- ( ( A = B /\ A =/= C ) -> B =/= C )

Syntax hints:   -> wi 4   /\ wa 102   = wceq 1284   =/= wne 2245

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 576  ax-in2 577  ax-5 1376  ax-gen 1378  ax-4 1440  ax-17 1459  ax-ext 2063

This theorem depends on definitions:  df-bi 115  df-cleq 2074  df-ne 2246

This theorem is referenced by:  pm13.181  2327