ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  neanior Unicode version

Theorem neanior 2372
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 2286 . . 3  |-  ( A  =/=  B  <->  -.  A  =  B )
2 df-ne 2286 . . 3  |-  ( C  =/=  D  <->  -.  C  =  D )
31, 2anbi12i 455 . 2  |-  ( ( A  =/=  B  /\  C  =/=  D )  <->  ( -.  A  =  B  /\  -.  C  =  D
) )
4 pm4.56 754 . 2  |-  ( ( -.  A  =  B  /\  -.  C  =  D )  <->  -.  ( A  =  B  \/  C  =  D )
)
53, 4bitri 183 1  |-  ( ( A  =/=  B  /\  C  =/=  D )  <->  -.  ( A  =  B  \/  C  =  D )
)
Colors of variables: wff set class
Syntax hints:   -. wn 3    /\ wa 103    <-> wb 104    \/ wo 682    = wceq 1316    =/= wne 2285
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 588  ax-in2 589  ax-io 683
This theorem depends on definitions:  df-bi 116  df-ne 2286
This theorem is referenced by:  nelpri  3521  nelprd  3523  0nelop  4140  lcmgcd  11686  lcmdvds  11687
  Copyright terms: Public domain W3C validator