Theorem neeq1 2380

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

Assertion

Ref	Expression
neeq1	|- ( A = B -> ( A =/= C <-> B =/= C ) )

Proof of Theorem neeq1

Step	Hyp	Ref	Expression
1		eqeq1 2203	. . 3 |- ( A = B -> ( A = C <-> B = C ) )
2	1	notbid 668	. 2 |- ( A = B -> ( -. A = C <-> -. B = C ) )
3		df-ne 2368	. 2 |- ( A =/= C <-> -. A = C )
4		df-ne 2368	. 2 |- ( B =/= C <-> -. B = C )
5	2, 3, 4	3bitr4g 223	1 |- ( A = B -> ( A =/= C <-> B =/= C ) )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 105   = wceq 1364   =/= wne 2367

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 1461  ax-gen 1463  ax-4 1524  ax-17 1540  ax-ext 2178

This theorem depends on definitions:  df-bi 117  df-cleq 2189  df-ne 2368

This theorem is referenced by:  neeq1i  2382  neeq1d  2385  nelrdva  2971  disji2  4027  0inp0  4200  frecabcl  6466  fiintim  7001  eldju2ndl  7147  updjudhf  7154  netap  7337  2oneel  7339  2omotaplemap  7340  2omotaplemst  7341  exmidapne  7343  xnn0nemnf  9340  uzn0  9634  xrnemnf  9869  xrnepnf  9870  ngtmnft  9909  xsubge0  9973  xposdif  9974  xleaddadd  9979  fztpval  10175  pcpre1  12486  pcqmul  12497  pcqcl  12500  xpsfrnel  13046  isnzr2  13816  fiinopn  14324  neapmkv  15799  neap0mkv  15800  ltlenmkv  15801