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Theorem pm4.54dc 841
Description: Theorem *4.54 of [WhiteheadRussell] p. 120, for decidable propositions. One form of DeMorgan's law. (Contributed by Jim Kingdon, 2-May-2018.)
Assertion
Ref Expression
pm4.54dc  |-  (DECID  ph  ->  (DECID  ps 
->  ( ( -.  ph  /\ 
ps )  <->  -.  ( ph  \/  -.  ps )
) ) )

Proof of Theorem pm4.54dc
StepHypRef Expression
1 dcn 782 . . . . 5  |-  (DECID  ph  -> DECID  -.  ph )
2 dfandc 814 . . . . 5  |-  (DECID  -.  ph  ->  (DECID  ps  ->  ( ( -.  ph  /\  ps )  <->  -.  ( -.  ph  ->  -. 
ps ) ) ) )
31, 2syl 14 . . . 4  |-  (DECID  ph  ->  (DECID  ps 
->  ( ( -.  ph  /\ 
ps )  <->  -.  ( -.  ph  ->  -.  ps )
) ) )
43imp 122 . . 3  |-  ( (DECID  ph  /\ DECID  ps )  ->  ( ( -. 
ph  /\  ps )  <->  -.  ( -.  ph  ->  -. 
ps ) ) )
5 pm4.66dc 838 . . . . 5  |-  (DECID  ph  ->  ( ( -.  ph  ->  -. 
ps )  <->  ( ph  \/  -.  ps ) ) )
65adantr 270 . . . 4  |-  ( (DECID  ph  /\ DECID  ps )  ->  ( ( -. 
ph  ->  -.  ps )  <->  (
ph  \/  -.  ps )
) )
76notbid 625 . . 3  |-  ( (DECID  ph  /\ DECID  ps )  ->  ( -.  ( -.  ph  ->  -.  ps )  <->  -.  ( ph  \/  -.  ps ) ) )
84, 7bitrd 186 . 2  |-  ( (DECID  ph  /\ DECID  ps )  ->  ( ( -. 
ph  /\  ps )  <->  -.  ( ph  \/  -.  ps ) ) )
98ex 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    \/ 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:  pm4.55dc  882
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