Theorem neeq2 2354

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 2180	. . 3 |- ( A = B -> ( C = A <-> C = B ) )
2	1	notbid 662	. 2 |- ( A = B -> ( -. C = A <-> -. C = B ) )
3		df-ne 2341	. 2 |- ( C =/= A <-> -. C = A )
4		df-ne 2341	. 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 1348   =/= wne 2340

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 609  ax-in2 610  ax-5 1440  ax-gen 1442  ax-4 1503  ax-17 1519  ax-ext 2152

This theorem depends on definitions:  df-bi 116  df-cleq 2163  df-ne 2341

This theorem is referenced by:  neeq2i  2356  neeq2d  2359  disji2  3982  fodjuomnilemdc  7120  xrlttri3  9754  neapmkv  14099