Theorem neanior 2427

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 2341	. . 3 |- ( A =/= B <-> -. A = B )
2		df-ne 2341	. . 3 |- ( C =/= D <-> -. C = D )
3	1, 2	anbi12i 457	. 2 |- ( ( A =/= B /\ C =/= D ) <-> ( -. A = B /\ -. C = D ) )
4		pm4.56 775	. 2 |- ( ( -. A = B /\ -. C = D ) <-> -. ( A = B \/ C = D ) )
5	3, 4	bitri 183	1 |- ( ( A =/= B /\ C =/= D ) <-> -. ( A = B \/ C = D ) )

Syntax hints:   -. wn 3   /\ wa 103   <-> wb 104   \/ wo 703   = 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-io 704

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

This theorem is referenced by:  nelpri  3607  nelprd  3609  eldifpr  3610  0nelop  4233  lcmgcd  12032  lcmdvds  12033  lgsdirnn0  13742  lgsdinn0  13743