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Theorem con2bidc 803
Description: Contraposition. (Contributed by Jim Kingdon, 17-Apr-2018.)
Assertion
Ref Expression
con2bidc  |-  (DECID  ph  ->  (DECID  ps 
->  ( ( ph  <->  -.  ps )  <->  ( ps  <->  -.  ph ) ) ) )

Proof of Theorem con2bidc
StepHypRef Expression
1 con1bidc 802 . . . . 5  |-  (DECID  ph  ->  (DECID  ps 
->  ( ( -.  ph  <->  ps )  <->  ( -.  ps  <->  ph ) ) ) )
21imp 122 . . . 4  |-  ( (DECID  ph  /\ DECID  ps )  ->  ( ( -. 
ph 
<->  ps )  <->  ( -.  ps 
<-> 
ph ) ) )
3 bicom 138 . . . 4  |-  ( ( -.  ph  <->  ps )  <->  ( ps  <->  -. 
ph ) )
4 bicom 138 . . . 4  |-  ( ( -.  ps  <->  ph )  <->  ( ph  <->  -. 
ps ) )
52, 3, 43bitr3g 220 . . 3  |-  ( (DECID  ph  /\ DECID  ps )  ->  ( ( ps  <->  -. 
ph )  <->  ( ph  <->  -. 
ps ) ) )
65bicomd 139 . 2  |-  ( (DECID  ph  /\ DECID  ps )  ->  ( ( ph  <->  -. 
ps )  <->  ( ps  <->  -. 
ph ) ) )
76ex 113 1  |-  (DECID  ph  ->  (DECID  ps 
->  ( ( ph  <->  -.  ps )  <->  ( ps  <->  -.  ph ) ) ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 102    <-> wb 103  DECID wdc 776
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 577  ax-in2 578  ax-io 663
This theorem depends on definitions:  df-bi 115  df-dc 777
This theorem is referenced by:  annimdc  879  pm4.55dc  880  orandc  881  nbbndc  1326
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