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  7413  updjudhf  7420  netap  7621  2oneel  7623  2omotaplemap  7624  2omotaplemst  7625  exmidapne  7627  xnn0nemnf  9646  uzn0  9948  xrnemnf  10190  xrnepnf  10191  ngtmnft  10230  xsubge0  10294  xposdif  10295  xleaddadd  10300  fztpval  10501  hashdmprop2dom  11311  fun2dmnop0  11317  pcpre1  13093  pcqmul  13104  pcqcl  13107  xpsfrnel  13716  isnzr2  14542  fiinopn  15157  umgrvad2edg  16574  isclwwlk  16757  eupth2lem2dc  16822  eupth2lem3lem6fi  16834  eupth2lem3lem4fi  16836  3dom  17140  pw1ndom3lem  17141  qdiff  17220  neapmkv  17240  neap0mkv  17241  ltlenmkv  17242