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

Theorem annimdc 879
Description: Express conjunction in terms of implication. The forward direction, annimim 816, is valid for all propositions, but as an equivalence, it requires a decidability condition. (Contributed by Jim Kingdon, 25-Apr-2018.)
Assertion
Ref Expression
annimdc  |-  (DECID  ph  ->  (DECID  ps 
->  ( ( ph  /\  -.  ps )  <->  -.  ( ph  ->  ps ) ) ) )

Proof of Theorem annimdc
StepHypRef Expression
1 imandc 820 . . . 4  |-  (DECID  ps  ->  ( ( ph  ->  ps ) 
<->  -.  ( ph  /\  -.  ps ) ) )
21adantl 271 . . 3  |-  ( (DECID  ph  /\ DECID  ps )  ->  ( ( ph  ->  ps )  <->  -.  ( ph  /\  -.  ps )
) )
3 dcim 818 . . . . 5  |-  (DECID  ph  ->  (DECID  ps 
-> DECID  ( ph  ->  ps )
) )
43imp 122 . . . 4  |-  ( (DECID  ph  /\ DECID  ps )  -> DECID 
( ph  ->  ps )
)
5 dcn 780 . . . . . 6  |-  (DECID  ps  -> DECID  -.  ps )
6 dcan 876 . . . . . 6  |-  (DECID  ph  ->  (DECID  -. 
ps  -> DECID 
( ph  /\  -.  ps ) ) )
75, 6syl5 32 . . . . 5  |-  (DECID  ph  ->  (DECID  ps 
-> DECID  ( ph  /\  -.  ps ) ) )
87imp 122 . . . 4  |-  ( (DECID  ph  /\ DECID  ps )  -> DECID 
( ph  /\  -.  ps ) )
9 con2bidc 803 . . . 4  |-  (DECID  ( ph  ->  ps )  ->  (DECID  ( ph  /\  -.  ps )  ->  ( ( ( ph  ->  ps )  <->  -.  ( ph  /\  -.  ps )
)  <->  ( ( ph  /\ 
-.  ps )  <->  -.  ( ph  ->  ps ) ) ) ) )
104, 8, 9sylc 61 . . 3  |-  ( (DECID  ph  /\ DECID  ps )  ->  ( ( (
ph  ->  ps )  <->  -.  ( ph  /\  -.  ps )
)  <->  ( ( ph  /\ 
-.  ps )  <->  -.  ( ph  ->  ps ) ) ) )
112, 10mpbid 145 . 2  |-  ( (DECID  ph  /\ DECID  ps )  ->  ( ( ph  /\ 
-.  ps )  <->  -.  ( ph  ->  ps ) ) )
1211ex 113 1  |-  (DECID  ph  ->  (DECID  ps 
->  ( ( ph  /\  -.  ps )  <->  -.  ( ph  ->  ps ) ) ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 102    <-> wb 103  DECID wdc 776
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
This theorem is referenced by:  xordidc  1331
  Copyright terms: Public domain W3C validator