Theorem neeq1 2380

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 2203	. . 3 |- ( A = B -> ( A = C <-> B = C ) )
2	1	notbid 668	. 2 |- ( A = B -> ( -. A = C <-> -. B = C ) )
3		df-ne 2368	. 2 |- ( A =/= C <-> -. A = C )
4		df-ne 2368	. 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 1364   =/= wne 2367

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 615  ax-in2 616  ax-5 1461  ax-gen 1463  ax-4 1524  ax-17 1540  ax-ext 2178

This theorem depends on definitions:  df-bi 117  df-cleq 2189  df-ne 2368

This theorem is referenced by:  neeq1i  2382  neeq1d  2385  nelrdva  2971  disji2  4026  0inp0  4199  frecabcl  6457  fiintim  6992  eldju2ndl  7138  updjudhf  7145  netap  7321  2oneel  7323  2omotaplemap  7324  2omotaplemst  7325  exmidapne  7327  xnn0nemnf  9323  uzn0  9617  xrnemnf  9852  xrnepnf  9853  ngtmnft  9892  xsubge0  9956  xposdif  9957  xleaddadd  9962  fztpval  10158  pcpre1  12461  pcqmul  12472  pcqcl  12475  xpsfrnel  12987  isnzr2  13740  fiinopn  14240  neapmkv  15712  neap0mkv  15713  ltlenmkv  15714