Theorem pm13.18 2426

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 2182	. . . 4 |- ( A = B -> ( A = C <-> B = C ) )
2	1	biimprd 158	. . 3 |- ( A = B -> ( B = C -> A = C ) )
3	2	necon3d 2389	. 2 |- ( A = B -> ( A =/= C -> B =/= C ) )
4	3	imp 124	1 |- ( ( A = B /\ A =/= C ) -> B =/= C )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1353   =/= wne 2345

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 614  ax-in2 615  ax-5 1445  ax-gen 1447  ax-4 1508  ax-17 1524  ax-ext 2157

This theorem depends on definitions:  df-bi 117  df-cleq 2168  df-ne 2346

This theorem is referenced by:  pm13.181  2427