Theorem neeq2 2392

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 2217	. . 3 |- ( A = B -> ( C = A <-> C = B ) )
2	1	notbid 669	. 2 |- ( A = B -> ( -. C = A <-> -. C = B ) )
3		df-ne 2379	. 2 |- ( C =/= A <-> -. C = A )
4		df-ne 2379	. 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 1373   =/= wne 2378

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 615  ax-in2 616  ax-5 1471  ax-gen 1473  ax-4 1534  ax-17 1550  ax-ext 2189

This theorem depends on definitions:  df-bi 117  df-cleq 2200  df-ne 2379

This theorem is referenced by:  neeq2i  2394  neeq2d  2397  disji2  4051  fodjuomnilemdc  7272  netap  7401  2oneel  7403  2omotaplemap  7404  2omotaplemst  7405  exmidapne  7407  xrlttri3  9954  hashdmprop2dom  11026  fun2dmnop0  11029  isnzr2  14061  neapmkv  16209  neap0mkv  16210  ltlenmkv  16211