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Theorem ne3anior 2394
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 2307 . . 3  |-  ( A  =/=  B  <->  -.  A  =  B )
2 df-ne 2307 . . 3  |-  ( C  =/=  D  <->  -.  C  =  D )
3 df-ne 2307 . . 3  |-  ( E  =/=  F  <->  -.  E  =  F )
41, 2, 33anbi123i 1170 . 2  |-  ( ( A  =/=  B  /\  C  =/=  D  /\  E  =/=  F )  <->  ( -.  A  =  B  /\  -.  C  =  D  /\  -.  E  =  F ) )
5 3ioran 977 . 2  |-  ( -.  ( A  =  B  \/  C  =  D  \/  E  =  F )  <->  ( -.  A  =  B  /\  -.  C  =  D  /\  -.  E  =  F ) )
64, 5bitr4i 186 1  |-  ( ( A  =/=  B  /\  C  =/=  D  /\  E  =/=  F )  <->  -.  ( A  =  B  \/  C  =  D  \/  E  =  F )
)
Colors of variables: wff set class
Syntax hints:   -. wn 3    <-> wb 104    \/ w3o 961    /\ w3a 962    = wceq 1331    =/= wne 2306
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 603  ax-in2 604  ax-io 698
This theorem depends on definitions:  df-bi 116  df-3or 963  df-3an 964  df-ne 2307
This theorem is referenced by: (None)
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