Theorem neeq2 2428

Description: Equality theorem for inequality. (Contributed by NM, 19-Nov-1994.)

Assertion

Ref	Expression
neeq2	|- ( A = B -> ( C =/= A <-> C =/= B ) )

Proof of Theorem neeq2

Step	Hyp	Ref	Expression
1		eqeq2 2244	. . 3 |- ( A = B -> ( C = A <-> C = B ) )
2	1	notbid 673	. 2 |- ( A = B -> ( -. C = A <-> -. C = B ) )
3		df-ne 2415	. 2 |- ( C =/= A <-> -. C = A )
4		df-ne 2415	. 2 |- ( C =/= B <-> -. C = B )
5	2, 3, 4	3bitr4g 223	1 |- ( A = B -> ( C =/= A <-> C =/= B ) )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 105   = wceq 1398   =/= wne 2414

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 619  ax-in2 620  ax-5 1496  ax-gen 1498  ax-4 1559  ax-17 1575  ax-ext 2216

This theorem depends on definitions:  df-bi 117  df-cleq 2227  df-ne 2415

This theorem is referenced by:  neeq2i  2430  neeq2d  2433  disji2  4106  fodjuomnilemdc  7448  netap  7584  2oneel  7586  2omotaplemap  7587  2omotaplemst  7588  exmidapne  7590  xrlttri3  10149  hashdmprop2dom  11241  fun2dmnop0  11247  isnzr2  14429  umgrvad2edg  16332  eupth2lem3lem4fi  16594  3dom  16888  qdiff  16959  neapmkv  16980  neap0mkv  16981  ltlenmkv  16982