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Theorem ne3anior 2339
Description: A De Morgan's law for inequality. (Contributed by NM, 30-Sep-2013.) (Proof rewritten by Jim Kingdon, 19-May-2018.)
Assertion
Ref Expression
ne3anior  |-  ( ( A  =/=  B  /\  C  =/=  D  /\  E  =/=  F )  <->  -.  ( A  =  B  \/  C  =  D  \/  E  =  F )
)

Proof of Theorem ne3anior
StepHypRef Expression
1 df-ne 2252 . . 3  |-  ( A  =/=  B  <->  -.  A  =  B )
2 df-ne 2252 . . 3  |-  ( C  =/=  D  <->  -.  C  =  D )
3 df-ne 2252 . . 3  |-  ( E  =/=  F  <->  -.  E  =  F )
41, 2, 33anbi123i 1130 . 2  |-  ( ( A  =/=  B  /\  C  =/=  D  /\  E  =/=  F )  <->  ( -.  A  =  B  /\  -.  C  =  D  /\  -.  E  =  F ) )
5 3ioran 937 . 2  |-  ( -.  ( A  =  B  \/  C  =  D  \/  E  =  F )  <->  ( -.  A  =  B  /\  -.  C  =  D  /\  -.  E  =  F ) )
64, 5bitr4i 185 1  |-  ( ( A  =/=  B  /\  C  =/=  D  /\  E  =/=  F )  <->  -.  ( A  =  B  \/  C  =  D  \/  E  =  F )
)
Colors of variables: wff set class
Syntax hints:   -. wn 3    <-> wb 103    \/ w3o 921    /\ w3a 922    = 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-3or 923  df-3an 924  df-ne 2252
This theorem is referenced by: (None)
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