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Theorem orandc 883
Description: Disjunction in terms of conjunction (De Morgan's law), for decidable propositions. Compare Theorem *4.57 of [WhiteheadRussell] p. 120. (Contributed by Jim Kingdon, 13-Dec-2021.)
Assertion
Ref Expression
orandc  |-  ( (DECID  ph  /\ DECID  ps )  ->  ( ( ph  \/  ps )  <->  -.  ( -.  ph  /\  -.  ps ) ) )

Proof of Theorem orandc
StepHypRef Expression
1 pm4.56 842 . 2  |-  ( ( -.  ph  /\  -.  ps ) 
<->  -.  ( ph  \/  ps ) )
2 dcn 782 . . . . 5  |-  (DECID  ph  -> DECID  -.  ph )
32adantr 270 . . . 4  |-  ( (DECID  ph  /\ DECID  ps )  -> DECID  -.  ph )
4 dcn 782 . . . . 5  |-  (DECID  ps  -> DECID  -.  ps )
54adantl 271 . . . 4  |-  ( (DECID  ph  /\ DECID  ps )  -> DECID  -.  ps )
6 dcan 878 . . . 4  |-  (DECID  -.  ph  ->  (DECID  -.  ps  -> DECID  ( -.  ph  /\  -.  ps ) ) )
73, 5, 6sylc 61 . . 3  |-  ( (DECID  ph  /\ DECID  ps )  -> DECID 
( -.  ph  /\  -.  ps ) )
8 dcor 879 . . . 4  |-  (DECID  ph  ->  (DECID  ps 
-> DECID  ( ph  \/  ps )
) )
98imp 122 . . 3  |-  ( (DECID  ph  /\ DECID  ps )  -> DECID 
( ph  \/  ps ) )
10 con2bidc 805 . . 3  |-  (DECID  ( -. 
ph  /\  -.  ps )  ->  (DECID  ( ph  \/  ps )  ->  ( ( ( -.  ph  /\  -.  ps ) 
<->  -.  ( ph  \/  ps ) )  <->  ( ( ph  \/  ps )  <->  -.  ( -.  ph  /\  -.  ps ) ) ) ) )
117, 9, 10sylc 61 . 2  |-  ( (DECID  ph  /\ DECID  ps )  ->  ( ( ( -.  ph  /\  -.  ps ) 
<->  -.  ( ph  \/  ps ) )  <->  ( ( ph  \/  ps )  <->  -.  ( -.  ph  /\  -.  ps ) ) ) )
121, 11mpbii 146 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    \/ wo 662  DECID wdc 778
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 779
This theorem is referenced by:  gcdaddm  10881
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