Theorem neeq2 2361

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 2187	. . 3 |- ( A = B -> ( C = A <-> C = B ) )
2	1	notbid 667	. 2 |- ( A = B -> ( -. C = A <-> -. C = B ) )
3		df-ne 2348	. 2 |- ( C =/= A <-> -. C = A )
4		df-ne 2348	. 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 1353   =/= wne 2347

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 614  ax-in2 615  ax-5 1447  ax-gen 1449  ax-4 1510  ax-17 1526  ax-ext 2159

This theorem depends on definitions:  df-bi 117  df-cleq 2170  df-ne 2348

This theorem is referenced by:  neeq2i  2363  neeq2d  2366  disji2  3998  fodjuomnilemdc  7144  netap  7255  2oneel  7257  2omotaplemap  7258  2omotaplemst  7259  exmidapne  7261  xrlttri3  9799  neapmkv  14901  neap0mkv  14902  ltlenmkv  14903