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Theorem dfbi3dc 1304
Description: An alternate definition of the biconditional for decidable propositions. Theorem *5.23 of [WhiteheadRussell] p. 124, but with decidability conditions. (Contributed by Jim Kingdon, 5-May-2018.)
Assertion
Ref Expression
dfbi3dc  |-  (DECID  ph  ->  (DECID  ps 
->  ( ( ph  <->  ps )  <->  ( ( ph  /\  ps )  \/  ( -.  ph 
/\  -.  ps )
) ) ) )

Proof of Theorem dfbi3dc
StepHypRef Expression
1 dcn 757 . . . 4  |-  (DECID  ps  -> DECID  -.  ps )
2 xordc 1299 . . . . 5  |-  (DECID  ph  ->  (DECID  -. 
ps  ->  ( -.  ( ph 
<->  -.  ps )  <->  ( ( ph  /\  -.  -.  ps )  \/  ( -.  ps  /\  -.  ph )
) ) ) )
32imp 119 . . . 4  |-  ( (DECID  ph  /\ DECID  -.  ps )  ->  ( -.  ( ph  <->  -.  ps )  <->  ( ( ph  /\  -.  -.  ps )  \/  ( -.  ps  /\  -.  ph ) ) ) )
41, 3sylan2 274 . . 3  |-  ( (DECID  ph  /\ DECID  ps )  ->  ( -.  ( ph 
<->  -.  ps )  <->  ( ( ph  /\  -.  -.  ps )  \/  ( -.  ps  /\  -.  ph )
) ) )
5 pm5.18dc 788 . . . 4  |-  (DECID  ph  ->  (DECID  ps 
->  ( ( ph  <->  ps )  <->  -.  ( ph  <->  -.  ps )
) ) )
65imp 119 . . 3  |-  ( (DECID  ph  /\ DECID  ps )  ->  ( ( ph  <->  ps )  <->  -.  ( ph  <->  -. 
ps ) ) )
7 notnotbdc 777 . . . . . 6  |-  (DECID  ps  ->  ( ps  <->  -.  -.  ps )
)
87anbi2d 445 . . . . 5  |-  (DECID  ps  ->  ( ( ph  /\  ps ) 
<->  ( ph  /\  -.  -.  ps ) ) )
9 ancom 257 . . . . . 6  |-  ( ( -.  ph  /\  -.  ps ) 
<->  ( -.  ps  /\  -.  ph ) )
109a1i 9 . . . . 5  |-  (DECID  ps  ->  ( ( -.  ph  /\  -.  ps )  <->  ( -.  ps  /\  -.  ph )
) )
118, 10orbi12d 717 . . . 4  |-  (DECID  ps  ->  ( ( ( ph  /\  ps )  \/  ( -.  ph  /\  -.  ps ) )  <->  ( ( ph  /\  -.  -.  ps )  \/  ( -.  ps  /\  -.  ph )
) ) )
1211adantl 266 . . 3  |-  ( (DECID  ph  /\ DECID  ps )  ->  ( ( (
ph  /\  ps )  \/  ( -.  ph  /\  -.  ps ) )  <->  ( ( ph  /\  -.  -.  ps )  \/  ( -.  ps  /\  -.  ph )
) ) )
134, 6, 123bitr4d 213 . 2  |-  ( (DECID  ph  /\ DECID  ps )  ->  ( ( ph  <->  ps )  <->  ( ( ph  /\ 
ps )  \/  ( -.  ph  /\  -.  ps ) ) ) )
1413ex 112 1  |-  (DECID  ph  ->  (DECID  ps 
->  ( ( ph  <->  ps )  <->  ( ( ph  /\  ps )  \/  ( -.  ph 
/\  -.  ps )
) ) ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 101    <-> wb 102    \/ wo 639  DECID wdc 753
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-in1 554  ax-in2 555  ax-io 640
This theorem depends on definitions:  df-bi 114  df-dc 754  df-xor 1283
This theorem is referenced by:  pm5.24dc  1305
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