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Theorem pm4.79dc 871
Description: Equivalence between a disjunction of two implications, and a conjunction and an implication. Based on theorem *4.79 of [WhiteheadRussell] p. 121 but with additional decidability antecedents. (Contributed by Jim Kingdon, 28-Mar-2018.)
Assertion
Ref Expression
pm4.79dc  |-  (DECID  ph  ->  (DECID  ps 
->  ( ( ( ps 
->  ph )  \/  ( ch  ->  ph ) )  <->  ( ( ps  /\  ch )  ->  ph ) ) ) )

Proof of Theorem pm4.79dc
StepHypRef Expression
1 id 19 . . . 4  |-  ( ( ps  ->  ph )  -> 
( ps  ->  ph )
)
2 id 19 . . . 4  |-  ( ( ch  ->  ph )  -> 
( ch  ->  ph )
)
31, 2jaoa 692 . . 3  |-  ( ( ( ps  ->  ph )  \/  ( ch  ->  ph )
)  ->  ( ( ps  /\  ch )  ->  ph ) )
4 simplimdc 828 . . . . . 6  |-  (DECID  ps  ->  ( -.  ( ps  ->  ph )  ->  ps )
)
5 pm3.3 259 . . . . . 6  |-  ( ( ( ps  /\  ch )  ->  ph )  ->  ( ps  ->  ( ch  ->  ph ) ) )
64, 5syl9 72 . . . . 5  |-  (DECID  ps  ->  ( ( ( ps  /\  ch )  ->  ph )  ->  ( -.  ( ps 
->  ph )  ->  ( ch  ->  ph ) ) ) )
7 dcim 809 . . . . . 6  |-  (DECID  ps  ->  (DECID  ph  -> DECID  ( ps  ->  ph ) ) )
8 pm2.54dc 859 . . . . . 6  |-  (DECID  ( ps 
->  ph )  ->  (
( -.  ( ps 
->  ph )  ->  ( ch  ->  ph ) )  -> 
( ( ps  ->  ph )  \/  ( ch 
->  ph ) ) ) )
97, 8syl6 33 . . . . 5  |-  (DECID  ps  ->  (DECID  ph  ->  ( ( -.  ( ps  ->  ph )  ->  ( ch  ->  ph ) )  -> 
( ( ps  ->  ph )  \/  ( ch 
->  ph ) ) ) ) )
106, 9syl5d 68 . . . 4  |-  (DECID  ps  ->  (DECID  ph  ->  ( ( ( ps 
/\  ch )  ->  ph )  ->  ( ( ps  ->  ph )  \/  ( ch 
->  ph ) ) ) ) )
1110imp 123 . . 3  |-  ( (DECID  ps 
/\ DECID  ph )  ->  ( ( ( ps  /\  ch )  ->  ph )  ->  (
( ps  ->  ph )  \/  ( ch  ->  ph )
) ) )
123, 11impbid2 142 . 2  |-  ( (DECID  ps 
/\ DECID  ph )  ->  ( ( ( ps  ->  ph )  \/  ( ch  ->  ph )
)  <->  ( ( ps 
/\  ch )  ->  ph )
) )
1312expcom 115 1  |-  (DECID  ph  ->  (DECID  ps 
->  ( ( ( ps 
->  ph )  \/  ( ch  ->  ph ) )  <->  ( ( ps  /\  ch )  ->  ph ) ) ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 103    <-> wb 104    \/ wo 680  DECID wdc 802
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 586  ax-in2 587  ax-io 681
This theorem depends on definitions:  df-bi 116  df-stab 799  df-dc 803
This theorem is referenced by: (None)
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