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Theorem con4biddc 825
Description: A contraposition deduction. (Contributed by Jim Kingdon, 18-May-2018.)
Hypothesis
Ref Expression
con4biddc.1  |-  ( ph  ->  (DECID  ps  ->  (DECID  ch  ->  ( -.  ps  <->  -.  ch )
) ) )
Assertion
Ref Expression
con4biddc  |-  ( ph  ->  (DECID  ps  ->  (DECID  ch  ->  ( ps  <->  ch ) ) ) )

Proof of Theorem con4biddc
StepHypRef Expression
1 con4biddc.1 . . . . . 6  |-  ( ph  ->  (DECID  ps  ->  (DECID  ch  ->  ( -.  ps  <->  -.  ch )
) ) )
2 bi2 129 . . . . . 6  |-  ( ( -.  ps  <->  -.  ch )  ->  ( -.  ch  ->  -. 
ps ) )
31, 2syl8 71 . . . . 5  |-  ( ph  ->  (DECID  ps  ->  (DECID  ch  ->  ( -.  ch  ->  -.  ps ) ) ) )
4 condc 821 . . . . . 6  |-  (DECID  ch  ->  ( ( -.  ch  ->  -. 
ps )  ->  ( ps  ->  ch ) ) )
54a2i 11 . . . . 5  |-  ( (DECID  ch 
->  ( -.  ch  ->  -. 
ps ) )  -> 
(DECID 
ch  ->  ( ps  ->  ch ) ) )
63, 5syl6 33 . . . 4  |-  ( ph  ->  (DECID  ps  ->  (DECID  ch  ->  ( ps  ->  ch )
) ) )
76imp31 254 . . 3  |-  ( ( ( ph  /\ DECID  ps )  /\ DECID  ch )  ->  ( ps  ->  ch ) )
8 bi1 117 . . . . . 6  |-  ( ( -.  ps  <->  -.  ch )  ->  ( -.  ps  ->  -. 
ch ) )
91, 8syl8 71 . . . . 5  |-  ( ph  ->  (DECID  ps  ->  (DECID  ch  ->  ( -.  ps  ->  -.  ch ) ) ) )
10 condc 821 . . . . . 6  |-  (DECID  ps  ->  ( ( -.  ps  ->  -. 
ch )  ->  ( ch  ->  ps ) ) )
1110imim2d 54 . . . . 5  |-  (DECID  ps  ->  ( (DECID  ch  ->  ( -.  ps  ->  -.  ch )
)  ->  (DECID  ch  ->  ( ch  ->  ps )
) ) )
129, 11sylcom 28 . . . 4  |-  ( ph  ->  (DECID  ps  ->  (DECID  ch  ->  ( ch  ->  ps )
) ) )
1312imp31 254 . . 3  |-  ( ( ( ph  /\ DECID  ps )  /\ DECID  ch )  ->  ( ch  ->  ps ) )
147, 13impbid 128 . 2  |-  ( ( ( ph  /\ DECID  ps )  /\ DECID  ch )  ->  ( ps  <->  ch ) )
1514exp31 359 1  |-  ( ph  ->  (DECID  ps  ->  (DECID  ch  ->  ( ps  <->  ch ) ) ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 103    <-> wb 104  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:  necon4abiddc  2356
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