Theorem neeq1 2433

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 2245	. . 3 |- ( A = B -> ( A = C <-> B = C ) )
2	1	notbid 677	. 2 |- ( A = B -> ( -. A = C <-> -. B = C ) )
3		df-ne 2421	. 2 |- ( A =/= C <-> -. A = C )
4		df-ne 2421	. 2 |- ( B =/= C <-> -. B = C )
5	2, 3, 4	3bitr4g 223	1 |- ( A = B -> ( A =/= C <-> B =/= C ) )

This proof depends on syntax axioms:   -. wn 3   -> wi 4   <-> wb 105   = wceq 1402   =/= wne 2420

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 623  ax-in2 624  ax-5 1500  ax-gen 1502  ax-4 1563  ax-17 1579  ax-ext 2220

This proof depends on definitions:  df-bi 117  df-cleq 2231  df-ne 2421

This theorem is used by:  neeq1i  2435  neeq1d  2438  nelrdva  3033  disji2  4122  0inp0  4305  frecabcl  6675  fiintim  7244  eldju2ndl  7418  updjudhf  7425  netap  7626  2oneel  7628  2omotaplemap  7629  2omotaplemst  7630  exmidapne  7632  xnn0nemnf  9650  uzn0  9953  xrnemnf  10195  xrnepnf  10196  ngtmnft  10235  xsubge0  10299  xposdif  10300  xleaddadd  10305  fztpval  10506  hashdmprop2dom  11317  fun2dmnop0  11323  pcpre1  13099  pcqmul  13110  pcqcl  13113  xpsfrnel  13723  isnzr2  14599  fiinopn  15247  umgrvad2edg  16674  isclwwlk  16857  eupth2lem2dc  16922  eupth2lem3lem6fi  16934  eupth2lem3lem4fi  16936  3dom  17240  pw1ndom3lem  17241  qdiff  17322  neapmkv  17342  neap0mkv  17343  ltlenmkv  17344