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Theorem ne3anior 2545
Description: A De Morgan's law for inequality. (Contributed by NM, 30-Sep-2013.)
Assertion
Ref Expression
ne3anior  |-  ( ( A  =/=  B  /\  C  =/=  D  /\  E  =/=  F )  <->  -.  ( A  =  B  \/  C  =  D  \/  E  =  F )
)

Proof of Theorem ne3anior
StepHypRef Expression
1 3anor 948 . 2  |-  ( ( A  =/=  B  /\  C  =/=  D  /\  E  =/=  F )  <->  -.  ( -.  A  =/=  B  \/  -.  C  =/=  D  \/  -.  E  =/=  F
) )
2 nne 2463 . . 3  |-  ( -.  A  =/=  B  <->  A  =  B )
3 nne 2463 . . 3  |-  ( -.  C  =/=  D  <->  C  =  D )
4 nne 2463 . . 3  |-  ( -.  E  =/=  F  <->  E  =  F )
52, 3, 43orbi123i 1141 . 2  |-  ( ( -.  A  =/=  B  \/  -.  C  =/=  D  \/  -.  E  =/=  F
)  <->  ( A  =  B  \/  C  =  D  \/  E  =  F ) )
61, 5xchbinx 301 1  |-  ( ( A  =/=  B  /\  C  =/=  D  /\  E  =/=  F )  <->  -.  ( A  =  B  \/  C  =  D  \/  E  =  F )
)
Colors of variables: wff set class
Syntax hints:   -. wn 3    <-> wb 176    \/ w3o 933    /\ w3a 934    = wceq 1632    =/= wne 2459
This theorem is referenced by:  eldiftp  23410  inttpemp  25242
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-ne 2461
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