Theorem neanior 2507

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 2421	. . 3 |- ( A =/= B <-> -. A = B )
2		df-ne 2421	. . 3 |- ( C =/= D <-> -. C = D )
3	1, 2	anbi12i 464	. 2 |- ( ( A =/= B /\ C =/= D ) <-> ( -. A = B /\ -. C = D ) )
4		pm4.56 792	. 2 |- ( ( -. A = B /\ -. C = D ) <-> -. ( A = B \/ C = D ) )
5	3, 4	bitri 184	1 |- ( ( A =/= B /\ C =/= D ) <-> -. ( A = B \/ C = D ) )

This proof depends on syntax axioms:   -. wn 3   /\ wa 104   <-> wb 105   \/ wo 720   = wceq 1402   =/= wne 2420

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 623  ax-in2 624  ax-io 721

This proof depends on definitions:  df-bi 117  df-ne 2421

This theorem is used by:  nelpri  3733  nelprd  3735  eldifpr  3736  0nelop  4388  lcmgcd  12856  lcmdvds  12857  domnmuln0  14582  lgsdirnn0  16166  lgsdinn0  16167  eupth2lem3lem7fi  16715