Theorem neeq2 2350

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 2175	. . 3 |- ( A = B -> ( C = A <-> C = B ) )
2	1	notbid 657	. 2 |- ( A = B -> ( -. C = A <-> -. C = B ) )
3		df-ne 2337	. 2 |- ( C =/= A <-> -. C = A )
4		df-ne 2337	. 2 |- ( C =/= B <-> -. C = B )
5	2, 3, 4	3bitr4g 222	1 |- ( A = B -> ( C =/= A <-> C =/= B ) )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 104   = wceq 1343   =/= wne 2336

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 604  ax-in2 605  ax-5 1435  ax-gen 1437  ax-4 1498  ax-17 1514  ax-ext 2147

This theorem depends on definitions:  df-bi 116  df-cleq 2158  df-ne 2337

This theorem is referenced by:  neeq2i  2352  neeq2d  2355  disji2  3975  fodjuomnilemdc  7108  xrlttri3  9733  neapmkv  13946