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Theorem pm4.63dc 780
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 778 . . . 4  |-  (DECID  ph  ->  (DECID  ps 
->  ( ( ph  /\  ps )  <->  -.  ( ph  ->  -.  ps ) ) ) )
21imp 115 . . 3  |-  ( (DECID  ph  /\ DECID  ps )  ->  ( ( ph  /\ 
ps )  <->  -.  ( ph  ->  -.  ps )
) )
32bicomd 129 . 2  |-  ( (DECID  ph  /\ DECID  ps )  ->  ( -.  ( ph  ->  -.  ps )  <->  (
ph  /\  ps )
) )
43ex 108 1  |-  (DECID  ph  ->  (DECID  ps 
->  ( -.  ( ph  ->  -.  ps )  <->  ( ph  /\ 
ps ) ) ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 97    <-> wb 98  DECID wdc 742
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-io 630
This theorem depends on definitions:  df-bi 110  df-dc 743
This theorem is referenced by:  pm4.67dc  781
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