Theorem neeq2 2417

Description: Equality theorem for inequality. (Contributed by NM, 19-Nov-1994.)

Assertion

Ref	Expression
neeq2	|- ( A = B -> ( C =/= A <-> C =/= B ) )

Proof of Theorem neeq2

Step	Hyp	Ref	Expression
1		eqeq2 2241	. . 3 |- ( A = B -> ( C = A <-> C = B ) )
2	1	notbid 673	. 2 |- ( A = B -> ( -. C = A <-> -. C = B ) )
3		df-ne 2404	. 2 |- ( C =/= A <-> -. C = A )
4		df-ne 2404	. 2 |- ( C =/= B <-> -. C = B )
5	2, 3, 4	3bitr4g 223	1 |- ( A = B -> ( C =/= A <-> C =/= B ) )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 105   = wceq 1398   =/= wne 2403

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 619  ax-in2 620  ax-5 1496  ax-gen 1498  ax-4 1559  ax-17 1575  ax-ext 2213

This theorem depends on definitions:  df-bi 117  df-cleq 2224  df-ne 2404

This theorem is referenced by:  neeq2i  2419  neeq2d  2422  disji2  4085  fodjuomnilemdc  7403  netap  7533  2oneel  7535  2omotaplemap  7536  2omotaplemst  7537  exmidapne  7539  xrlttri3  10093  hashdmprop2dom  11171  fun2dmnop0  11177  isnzr2  14279  umgrvad2edg  16152  eupth2lem3lem4fi  16414  3dom  16708  qdiff  16781  neapmkv  16801  neap0mkv  16802  ltlenmkv  16803