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Theorem nebidc 2329
Description: Contraposition law for inequality. (Contributed by Jim Kingdon, 19-May-2018.)
Assertion
Ref Expression
nebidc  |-  (DECID  A  =  B  ->  (DECID  C  =  D  ->  ( ( A  =  B  <->  C  =  D )  <->  ( A  =/=  B  <->  C  =/=  D
) ) ) )

Proof of Theorem nebidc
StepHypRef Expression
1 id 19 . . . 4  |-  ( ( A  =  B  <->  C  =  D )  ->  ( A  =  B  <->  C  =  D ) )
21necon3bid 2290 . . 3  |-  ( ( A  =  B  <->  C  =  D )  ->  ( A  =/=  B  <->  C  =/=  D ) )
3 id 19 . . . . . . . 8  |-  ( ( A  =/=  B  <->  C  =/=  D )  ->  ( A  =/=  B  <->  C  =/=  D
) )
43a1d 22 . . . . . . 7  |-  ( ( A  =/=  B  <->  C  =/=  D )  ->  (DECID  C  =  D  ->  ( A  =/= 
B  <->  C  =/=  D
) ) )
54a1d 22 . . . . . 6  |-  ( ( A  =/=  B  <->  C  =/=  D )  ->  (DECID  A  =  B  ->  (DECID  C  =  D  -> 
( A  =/=  B  <->  C  =/=  D ) ) ) )
65necon4biddc 2324 . . . . 5  |-  ( ( A  =/=  B  <->  C  =/=  D )  ->  (DECID  A  =  B  ->  (DECID  C  =  D  -> 
( A  =  B  <-> 
C  =  D ) ) ) )
76com3l 80 . . . 4  |-  (DECID  A  =  B  ->  (DECID  C  =  D  ->  ( ( A  =/=  B  <->  C  =/=  D )  ->  ( A  =  B  <->  C  =  D
) ) ) )
87imp 122 . . 3  |-  ( (DECID  A  =  B  /\ DECID  C  =  D )  ->  (
( A  =/=  B  <->  C  =/=  D )  -> 
( A  =  B  <-> 
C  =  D ) ) )
92, 8impbid2 141 . 2  |-  ( (DECID  A  =  B  /\ DECID  C  =  D )  ->  (
( A  =  B  <-> 
C  =  D )  <-> 
( A  =/=  B  <->  C  =/=  D ) ) )
109ex 113 1  |-  (DECID  A  =  B  ->  (DECID  C  =  D  ->  ( ( A  =  B  <->  C  =  D )  <->  ( A  =/=  B  <->  C  =/=  D
) ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 102    <-> wb 103  DECID wdc 776    = wceq 1285    =/= wne 2249
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-dc 777  df-ne 2250
This theorem is referenced by:  rpexp  10739
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