Theorem pm13.181 2327

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 2083	. 2 |- ( A = B <-> B = A )
2		pm13.18 2326	. 2 |- ( ( B = A /\ B =/= C ) -> A =/= C )
3	1, 2	sylanb 278	1 |- ( ( A = B /\ B =/= C ) -> A =/= 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:  fzprval  9099  mod2eq1n2dvds  10279