Theorem neeq1 2416

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 2238	. . 3 |- ( A = B -> ( A = C <-> B = C ) )
2	1	notbid 673	. 2 |- ( A = B -> ( -. A = C <-> -. B = C ) )
3		df-ne 2404	. 2 |- ( A =/= C <-> -. A = C )
4		df-ne 2404	. 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 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:  neeq1i  2418  neeq1d  2421  nelrdva  3014  disji2  4085  0inp0  4262  frecabcl  6608  fiintim  7166  eldju2ndl  7331  updjudhf  7338  netap  7533  2oneel  7535  2omotaplemap  7536  2omotaplemst  7537  exmidapne  7539  xnn0nemnf  9537  uzn0  9833  xrnemnf  10073  xrnepnf  10074  ngtmnft  10113  xsubge0  10177  xposdif  10178  xleaddadd  10183  fztpval  10380  hashdmprop2dom  11171  fun2dmnop0  11177  pcpre1  12945  pcqmul  12956  pcqcl  12959  xpsfrnel  13507  isnzr2  14279  fiinopn  14815  umgrvad2edg  16152  isclwwlk  16335  eupth2lem2dc  16400  eupth2lem3lem6fi  16412  eupth2lem3lem4fi  16414  3dom  16708  pw1ndom3lem  16709  qdiff  16781  neapmkv  16801  neap0mkv  16802  ltlenmkv  16803