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Theorem neanior 2338
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
StepHypRef Expression
1 df-ne 2252 . . 3  |-  ( A  =/=  B  <->  -.  A  =  B )
2 df-ne 2252 . . 3  |-  ( C  =/=  D  <->  -.  C  =  D )
31, 2anbi12i 448 . 2  |-  ( ( A  =/=  B  /\  C  =/=  D )  <->  ( -.  A  =  B  /\  -.  C  =  D
) )
4 pm4.56 842 . 2  |-  ( ( -.  A  =  B  /\  -.  C  =  D )  <->  -.  ( A  =  B  \/  C  =  D )
)
53, 4bitri 182 1  |-  ( ( A  =/=  B  /\  C  =/=  D )  <->  -.  ( A  =  B  \/  C  =  D )
)
Colors of variables: wff set class
Syntax hints:   -. wn 3    /\ wa 102    <-> wb 103    \/ wo 662    = wceq 1287    =/= wne 2251
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 577  ax-in2 578  ax-io 663
This theorem depends on definitions:  df-bi 115  df-ne 2252
This theorem is referenced by:  nelpri  3455  0nelop  4049  lcmgcd  10935  lcmdvds  10936
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