Theorem neeq2 2378

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 2203	. . 3 |- ( A = B -> ( C = A <-> C = B ) )
2	1	notbid 668	. 2 |- ( A = B -> ( -. C = A <-> -. C = B ) )
3		df-ne 2365	. 2 |- ( C =/= A <-> -. C = A )
4		df-ne 2365	. 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 1364   =/= wne 2364

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 1458  ax-gen 1460  ax-4 1521  ax-17 1537  ax-ext 2175

This theorem depends on definitions:  df-bi 117  df-cleq 2186  df-ne 2365

This theorem is referenced by:  neeq2i  2380  neeq2d  2383  disji2  4022  fodjuomnilemdc  7203  netap  7314  2oneel  7316  2omotaplemap  7317  2omotaplemst  7318  exmidapne  7320  xrlttri3  9863  isnzr2  13680  neapmkv  15563  neap0mkv  15564  ltlenmkv  15565