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  4303  frecabcl  6670  fiintim  7238  eldju2ndl  7412  updjudhf  7419  netap  7620  2oneel  7622  2omotaplemap  7623  2omotaplemst  7624  exmidapne  7626  xnn0nemnf  9641  uzn0  9938  xrnemnf  10179  xrnepnf  10180  ngtmnft  10219  xsubge0  10283  xposdif  10284  xleaddadd  10289  fztpval  10490  hashdmprop2dom  11296  fun2dmnop0  11302  pcpre1  13071  pcqmul  13082  pcqcl  13085  xpsfrnel  13665  isnzr2  14491  fiinopn  15105  umgrvad2edg  16452  isclwwlk  16635  eupth2lem2dc  16700  eupth2lem3lem6fi  16712  eupth2lem3lem4fi  16714  3dom  17018  pw1ndom3lem  17019  qdiff  17098  neapmkv  17118  neap0mkv  17119  ltlenmkv  17120