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Theorem nbbndc 1285
Description: Move negation outside of biconditional, for decidable propositions. Compare Theorem *5.18 of [WhiteheadRussell] p. 124. (Contributed by Jim Kingdon, 18-Apr-2018.)
Assertion
Ref Expression
nbbndc  |-  (DECID  ph  ->  (DECID  ps 
->  ( ( -.  ph  <->  ps )  <->  -.  ( ph  <->  ps ) ) ) )

Proof of Theorem nbbndc
StepHypRef Expression
1 xor3dc 1278 . . . . 5  |-  (DECID  ph  ->  (DECID  ps 
->  ( -.  ( ph  <->  ps )  <->  ( ph  <->  -.  ps )
) ) )
21imp 115 . . . 4  |-  ( (DECID  ph  /\ DECID  ps )  ->  ( -.  ( ph 
<->  ps )  <->  ( ph  <->  -. 
ps ) ) )
3 con2bidc 769 . . . . 5  |-  (DECID  ph  ->  (DECID  ps 
->  ( ( ph  <->  -.  ps )  <->  ( ps  <->  -.  ph ) ) ) )
43imp 115 . . . 4  |-  ( (DECID  ph  /\ DECID  ps )  ->  ( ( ph  <->  -. 
ps )  <->  ( ps  <->  -. 
ph ) ) )
52, 4bitrd 177 . . 3  |-  ( (DECID  ph  /\ DECID  ps )  ->  ( -.  ( ph 
<->  ps )  <->  ( ps  <->  -. 
ph ) ) )
6 bicom 128 . . 3  |-  ( ( ps  <->  -.  ph )  <->  ( -.  ph  <->  ps ) )
75, 6syl6rbb 186 . 2  |-  ( (DECID  ph  /\ DECID  ps )  ->  ( ( -. 
ph 
<->  ps )  <->  -.  ( ph 
<->  ps ) ) )
87ex 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:  biassdc  1286
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