Theorem pm13.181 2409

Description: Theorem *13.181 in [WhiteheadRussell] p. 178. (Contributed by Andrew Salmon, 3-Jun-2011.)

Assertion

Ref	Expression
pm13.181	|- ( ( A = B /\ B =/= C ) -> A =/= C )

Proof of Theorem pm13.181

Step	Hyp	Ref	Expression
1		eqcom 2159	. 2 |- ( A = B <-> B = A )
2		pm13.18 2408	. 2 |- ( ( B = A /\ B =/= C ) -> A =/= C )
3	1, 2	sylanb 282	1 |- ( ( A = B /\ B =/= C ) -> A =/= C )

Syntax hints:   -> wi 4   /\ wa 103   = wceq 1335   =/= wne 2327

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 604  ax-in2 605  ax-5 1427  ax-gen 1429  ax-4 1490  ax-17 1506  ax-ext 2139

This theorem depends on definitions:  df-bi 116  df-cleq 2150  df-ne 2328

This theorem is referenced by:  nninfisollemne  7074  fzprval  9984  mod2eq1n2dvds  11769