Theorem neeq1 2413

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 2236	. . 3 |- ( A = B -> ( A = C <-> B = C ) )
2	1	notbid 671	. 2 |- ( A = B -> ( -. A = C <-> -. B = C ) )
3		df-ne 2401	. 2 |- ( A =/= C <-> -. A = C )
4		df-ne 2401	. 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 1395   =/= wne 2400

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 617  ax-in2 618  ax-5 1493  ax-gen 1495  ax-4 1556  ax-17 1572  ax-ext 2211

This theorem depends on definitions:  df-bi 117  df-cleq 2222  df-ne 2401

This theorem is referenced by:  neeq1i  2415  neeq1d  2418  nelrdva  3010  disji2  4075  0inp0  4250  frecabcl  6551  fiintim  7104  eldju2ndl  7250  updjudhf  7257  netap  7451  2oneel  7453  2omotaplemap  7454  2omotaplemst  7455  exmidapne  7457  xnn0nemnf  9454  uzn0  9750  xrnemnf  9985  xrnepnf  9986  ngtmnft  10025  xsubge0  10089  xposdif  10090  xleaddadd  10095  fztpval  10291  hashdmprop2dom  11079  fun2dmnop0  11082  pcpre1  12831  pcqmul  12842  pcqcl  12845  xpsfrnel  13393  isnzr2  14164  fiinopn  14694  umgrvad2edg  16025  isclwwlk  16137  3dom  16439  pw1ndom3lem  16440  neapmkv  16524  neap0mkv  16525  ltlenmkv  16526