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Theorem annimdc 904
Description: Express conjunction in terms of implication. The forward direction, annimim 658, 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 857 . . . 4  |-  (DECID  ps  ->  ( ( ph  ->  ps ) 
<->  -.  ( ph  /\  -.  ps ) ) )
21adantl 273 . . 3  |-  ( (DECID  ph  /\ DECID  ps )  ->  ( ( ph  ->  ps )  <->  -.  ( ph  /\  -.  ps )
) )
3 dcim 809 . . . . 5  |-  (DECID  ph  ->  (DECID  ps 
-> DECID  ( ph  ->  ps )
) )
43imp 123 . . . 4  |-  ( (DECID  ph  /\ DECID  ps )  -> DECID 
( ph  ->  ps )
)
5 dcn 810 . . . . . 6  |-  (DECID  ps  -> DECID  -.  ps )
6 dcan 901 . . . . . 6  |-  (DECID  ph  ->  (DECID  -. 
ps  -> DECID 
( ph  /\  -.  ps ) ) )
75, 6syl5 32 . . . . 5  |-  (DECID  ph  ->  (DECID  ps 
-> DECID  ( ph  /\  -.  ps ) ) )
87imp 123 . . . 4  |-  ( (DECID  ph  /\ DECID  ps )  -> DECID 
( ph  /\  -.  ps ) )
9 con2bidc 843 . . . 4  |-  (DECID  ( ph  ->  ps )  ->  (DECID  ( ph  /\  -.  ps )  ->  ( ( ( ph  ->  ps )  <->  -.  ( ph  /\  -.  ps )
)  <->  ( ( ph  /\ 
-.  ps )  <->  -.  ( ph  ->  ps ) ) ) ) )
104, 8, 9sylc 62 . . 3  |-  ( (DECID  ph  /\ DECID  ps )  ->  ( ( (
ph  ->  ps )  <->  -.  ( ph  /\  -.  ps )
)  <->  ( ( ph  /\ 
-.  ps )  <->  -.  ( ph  ->  ps ) ) ) )
112, 10mpbid 146 . 2  |-  ( (DECID  ph  /\ DECID  ps )  ->  ( ( ph  /\ 
-.  ps )  <->  -.  ( ph  ->  ps ) ) )
1211ex 114 1  |-  (DECID  ph  ->  (DECID  ps 
->  ( ( ph  /\  -.  ps )  <->  -.  ( ph  ->  ps ) ) ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 103    <-> wb 104  DECID wdc 802
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 586  ax-in2 587  ax-io 681
This theorem depends on definitions:  df-bi 116  df-stab 799  df-dc 803
This theorem is referenced by:  xordidc  1360
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