Theorem neanior 2434

Description: A De Morgan's law for inequality. (Contributed by NM, 18-May-2007.)

Assertion

Ref	Expression
neanior	|- ( ( A =/= B /\ C =/= D ) <-> -. ( A = B \/ C = D ) )

Proof of Theorem neanior

Step	Hyp	Ref	Expression
1		df-ne 2348	. . 3 |- ( A =/= B <-> -. A = B )
2		df-ne 2348	. . 3 |- ( C =/= D <-> -. C = D )
3	1, 2	anbi12i 460	. 2 |- ( ( A =/= B /\ C =/= D ) <-> ( -. A = B /\ -. C = D ) )
4		pm4.56 780	. 2 |- ( ( -. A = B /\ -. C = D ) <-> -. ( A = B \/ C = D ) )
5	3, 4	bitri 184	1 |- ( ( A =/= B /\ C =/= D ) <-> -. ( A = B \/ C = D ) )

Syntax hints:   -. wn 3   /\ wa 104   <-> wb 105   \/ wo 708   = wceq 1353   =/= wne 2347

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 614  ax-in2 615  ax-io 709

This theorem depends on definitions:  df-bi 117  df-ne 2348

This theorem is referenced by:  nelpri  3615  nelprd  3617  eldifpr  3618  0nelop  4245  lcmgcd  12061  lcmdvds  12062  lgsdirnn0  14115  lgsdinn0  14116