Theorem pm13.18 2875

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 2626	. . . 4 |- ( A = B -> ( A = C <-> B = C ) )
2	1	biimprd 238	. . 3 |- ( A = B -> ( B = C -> A = C ) )
3	2	necon3d 2815	. 2 |- ( A = B -> ( A =/= C -> B =/= C ) )
4	3	imp 445	1 |- ( ( A = B /\ A =/= C ) -> B =/= C )

Syntax hints:   -> wi 4   /\ wa 384   = wceq 1483   =/= wne 2794

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-an 386  df-ex 1705  df-cleq 2615  df-ne 2795

This theorem is referenced by:  pm13.181  2876  frgrwopreglem5a  27175  4atexlemex4  35359  cncfiooicclem1  40106