Theorem pm13.18 2421

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 2177	. . . 4 |- ( A = B -> ( A = C <-> B = C ) )
2	1	biimprd 157	. . 3 |- ( A = B -> ( B = C -> A = C ) )
3	2	necon3d 2384	. 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 1348   =/= wne 2340

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 609  ax-in2 610  ax-5 1440  ax-gen 1442  ax-4 1503  ax-17 1519  ax-ext 2152

This theorem depends on definitions:  df-bi 116  df-cleq 2163  df-ne 2341

This theorem is referenced by:  pm13.181  2422