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

Theorem nebidc 2365
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 2326 . . 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 2360 . . . . 5  |-  ( ( A  =/=  B  <->  C  =/=  D )  ->  (DECID  A  =  B  ->  (DECID  C  =  D  -> 
( A  =  B  <-> 
C  =  D ) ) ) )
76com3l 81 . . . 4  |-  (DECID  A  =  B  ->  (DECID  C  =  D  ->  ( ( A  =/=  B  <->  C  =/=  D )  ->  ( A  =  B  <->  C  =  D
) ) ) )
87imp 123 . . 3  |-  ( (DECID  A  =  B  /\ DECID  C  =  D )  ->  (
( A  =/=  B  <->  C  =/=  D )  -> 
( A  =  B  <-> 
C  =  D ) ) )
92, 8impbid2 142 . 2  |-  ( (DECID  A  =  B  /\ DECID  C  =  D )  ->  (
( A  =  B  <-> 
C  =  D )  <-> 
( A  =/=  B  <->  C  =/=  D ) ) )
109ex 114 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 103    <-> wb 104  DECID wdc 804    = 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-stab 801  df-dc 805  df-ne 2286
This theorem is referenced by:  rpexp  11758
  Copyright terms: Public domain W3C validator