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Theorem pm4.63dc 856
Description: Theorem *4.63 of [WhiteheadRussell] p. 120, for decidable propositions. (Contributed by Jim Kingdon, 1-May-2018.)
Assertion
Ref Expression
pm4.63dc  |-  (DECID  ph  ->  (DECID  ps 
->  ( -.  ( ph  ->  -.  ps )  <->  ( ph  /\ 
ps ) ) ) )

Proof of Theorem pm4.63dc
StepHypRef Expression
1 dfandc 854 . . . 4  |-  (DECID  ph  ->  (DECID  ps 
->  ( ( ph  /\  ps )  <->  -.  ( ph  ->  -.  ps ) ) ) )
21imp 123 . . 3  |-  ( (DECID  ph  /\ DECID  ps )  ->  ( ( ph  /\ 
ps )  <->  -.  ( ph  ->  -.  ps )
) )
32bicomd 140 . 2  |-  ( (DECID  ph  /\ DECID  ps )  ->  ( -.  ( ph  ->  -.  ps )  <->  (
ph  /\  ps )
) )
43ex 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 804
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 588  ax-in2 589  ax-io 683
This theorem depends on definitions:  df-bi 116  df-stab 801  df-dc 805
This theorem is referenced by:  pm4.67dc  857
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