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Theorem orandc 941
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 pm4.56 781 . 2  |-  ( ( -.  ph  /\  -.  ps ) 
<->  -.  ( ph  \/  ps ) )
2 dcn 843 . . . 4  |-  (DECID  ph  -> DECID  -.  ph )
3 dcn 843 . . . 4  |-  (DECID  ps  -> DECID  -.  ps )
4 dcan 935 . . . 4  |-  ( (DECID  -. 
ph  /\ DECID  -.  ps )  -> DECID  ( -.  ph  /\  -.  ps ) )
52, 3, 4syl2an 289 . . 3  |-  ( (DECID  ph  /\ DECID  ps )  -> DECID 
( -.  ph  /\  -.  ps ) )
6 dcor 937 . . . 4  |-  (DECID  ph  ->  (DECID  ps 
-> DECID  ( ph  \/  ps )
) )
76imp 124 . . 3  |-  ( (DECID  ph  /\ DECID  ps )  -> DECID 
( ph  \/  ps ) )
8 con2bidc 876 . . 3  |-  (DECID  ( -. 
ph  /\  -.  ps )  ->  (DECID  ( ph  \/  ps )  ->  ( ( ( -.  ph  /\  -.  ps ) 
<->  -.  ( ph  \/  ps ) )  <->  ( ( ph  \/  ps )  <->  -.  ( -.  ph  /\  -.  ps ) ) ) ) )
95, 7, 8sylc 62 . 2  |-  ( (DECID  ph  /\ DECID  ps )  ->  ( ( ( -.  ph  /\  -.  ps ) 
<->  -.  ( ph  \/  ps ) )  <->  ( ( ph  \/  ps )  <->  -.  ( -.  ph  /\  -.  ps ) ) ) )
101, 9mpbii 148 1  |-  ( (DECID  ph  /\ DECID  ps )  ->  ( ( ph  \/  ps )  <->  -.  ( -.  ph  /\  -.  ps ) ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 104    <-> wb 105    \/ wo 709  DECID wdc 835
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 710
This theorem depends on definitions:  df-bi 117  df-stab 832  df-dc 836
This theorem is referenced by:  gcdaddm  12111
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