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Theorem pm4.55dc 938
Description: Theorem *4.55 of [WhiteheadRussell] p. 120, for decidable propositions. (Contributed by Jim Kingdon, 2-May-2018.)
Assertion
Ref Expression
pm4.55dc  |-  (DECID  ph  ->  (DECID  ps 
->  ( -.  ( -. 
ph  /\  ps )  <->  (
ph  \/  -.  ps )
) ) )

Proof of Theorem pm4.55dc
StepHypRef Expression
1 pm4.54dc 902 . . . . 5  |-  (DECID  ph  ->  (DECID  ps 
->  ( ( -.  ph  /\ 
ps )  <->  -.  ( ph  \/  -.  ps )
) ) )
21imp 124 . . . 4  |-  ( (DECID  ph  /\ DECID  ps )  ->  ( ( -. 
ph  /\  ps )  <->  -.  ( ph  \/  -.  ps ) ) )
3 dcn 842 . . . . . . . . 9  |-  (DECID  ps  -> DECID  -.  ps )
43anim2i 342 . . . . . . . 8  |-  ( (DECID  ph  /\ DECID  ps )  ->  (DECID 
ph  /\ DECID  -.  ps ) )
5 dcor 935 . . . . . . . . 9  |-  (DECID  ph  ->  (DECID  -. 
ps  -> DECID 
( ph  \/  -.  ps ) ) )
65imp 124 . . . . . . . 8  |-  ( (DECID  ph  /\ DECID  -.  ps )  -> DECID  ( ph  \/  -.  ps ) )
74, 6syl 14 . . . . . . 7  |-  ( (DECID  ph  /\ DECID  ps )  -> DECID 
( ph  \/  -.  ps ) )
8 dcn 842 . . . . . . . . 9  |-  (DECID  ph  -> DECID  -.  ph )
9 dcan2 934 . . . . . . . . 9  |-  (DECID  -.  ph  ->  (DECID  ps  -> DECID  ( -.  ph  /\  ps ) ) )
108, 9syl 14 . . . . . . . 8  |-  (DECID  ph  ->  (DECID  ps 
-> DECID  ( -.  ph  /\  ps ) ) )
1110imp 124 . . . . . . 7  |-  ( (DECID  ph  /\ DECID  ps )  -> DECID 
( -.  ph  /\  ps ) )
127, 11jca 306 . . . . . 6  |-  ( (DECID  ph  /\ DECID  ps )  ->  (DECID  ( ph  \/  -.  ps )  /\ DECID  ( -.  ph  /\  ps ) ) )
13 con2bidc 875 . . . . . . 7  |-  (DECID  ( ph  \/  -.  ps )  -> 
(DECID  ( -.  ph  /\  ps )  ->  ( ( ( ph  \/  -.  ps )  <->  -.  ( -.  ph 
/\  ps ) )  <->  ( ( -.  ph  /\  ps )  <->  -.  ( ph  \/  -.  ps ) ) ) ) )
1413imp 124 . . . . . 6  |-  ( (DECID  (
ph  \/  -.  ps )  /\ DECID  ( -.  ph  /\  ps ) )  ->  (
( ( ph  \/  -.  ps )  <->  -.  ( -.  ph  /\  ps )
)  <->  ( ( -. 
ph  /\  ps )  <->  -.  ( ph  \/  -.  ps ) ) ) )
1512, 14syl 14 . . . . 5  |-  ( (DECID  ph  /\ DECID  ps )  ->  ( ( (
ph  \/  -.  ps )  <->  -.  ( -.  ph  /\  ps ) )  <->  ( ( -.  ph  /\  ps )  <->  -.  ( ph  \/  -.  ps ) ) ) )
1615biimprd 158 . . . 4  |-  ( (DECID  ph  /\ DECID  ps )  ->  ( ( ( -.  ph  /\  ps )  <->  -.  ( ph  \/  -.  ps ) )  ->  (
( ph  \/  -.  ps )  <->  -.  ( -.  ph 
/\  ps ) ) ) )
172, 16mpd 13 . . 3  |-  ( (DECID  ph  /\ DECID  ps )  ->  ( ( ph  \/  -.  ps )  <->  -.  ( -.  ph  /\  ps )
) )
1817bicomd 141 . 2  |-  ( (DECID  ph  /\ DECID  ps )  ->  ( -.  ( -.  ph  /\  ps )  <->  (
ph  \/  -.  ps )
) )
1918ex 115 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 708  DECID wdc 834
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 614  ax-in2 615  ax-io 709
This theorem depends on definitions:  df-bi 117  df-stab 831  df-dc 835
This theorem is referenced by: (None)
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