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Theorem pm5.18dc 811
Description: Relationship between an equivalence and an equivalence with some negation, for decidable propositions. Based on theorem *5.18 of [WhiteheadRussell] p. 124. Given decidability, we can consider  -.  ( ph  <->  -.  ps ) to represent "negated exclusive-or". (Contributed by Jim Kingdon, 4-Apr-2018.)
Assertion
Ref Expression
pm5.18dc  |-  (DECID  ph  ->  (DECID  ps 
->  ( ( ph  <->  ps )  <->  -.  ( ph  <->  -.  ps )
) ) )

Proof of Theorem pm5.18dc
StepHypRef Expression
1 df-dc 777 . 2  |-  (DECID  ph  <->  ( ph  \/  -.  ph ) )
2 pm5.501 242 . . . . . . . 8  |-  ( ph  ->  ( -.  ps  <->  ( ph  <->  -. 
ps ) ) )
32a1d 22 . . . . . . 7  |-  ( ph  ->  (DECID  ps  ->  ( -.  ps 
<->  ( ph  <->  -.  ps )
) ) )
43con1biddc 804 . . . . . 6  |-  ( ph  ->  (DECID  ps  ->  ( -.  ( ph  <->  -.  ps )  <->  ps ) ) )
54imp 122 . . . . 5  |-  ( (
ph  /\ DECID  ps )  ->  ( -.  ( ph  <->  -.  ps )  <->  ps ) )
6 pm5.501 242 . . . . . 6  |-  ( ph  ->  ( ps  <->  ( ph  <->  ps ) ) )
76adantr 270 . . . . 5  |-  ( (
ph  /\ DECID  ps )  ->  ( ps 
<->  ( ph  <->  ps )
) )
85, 7bitr2d 187 . . . 4  |-  ( (
ph  /\ DECID  ps )  ->  (
( ph  <->  ps )  <->  -.  ( ph 
<->  -.  ps ) ) )
98ex 113 . . 3  |-  ( ph  ->  (DECID  ps  ->  ( ( ph 
<->  ps )  <->  -.  ( ph 
<->  -.  ps ) ) ) )
10 dcn 780 . . . . . . 7  |-  (DECID  ps  -> DECID  -.  ps )
11 nbn2 646 . . . . . . . . 9  |-  ( -. 
ph  ->  ( -.  -.  ps 
<->  ( ph  <->  -.  ps )
) )
1211a1d 22 . . . . . . . 8  |-  ( -. 
ph  ->  (DECID  -.  ps  ->  ( -.  -.  ps  <->  ( ph  <->  -. 
ps ) ) ) )
1312con1biddc 804 . . . . . . 7  |-  ( -. 
ph  ->  (DECID  -.  ps  ->  ( -.  ( ph  <->  -.  ps )  <->  -. 
ps ) ) )
1410, 13syl5 32 . . . . . 6  |-  ( -. 
ph  ->  (DECID  ps  ->  ( -.  ( ph  <->  -.  ps )  <->  -. 
ps ) ) )
1514imp 122 . . . . 5  |-  ( ( -.  ph  /\ DECID  ps )  ->  ( -.  ( ph  <->  -.  ps )  <->  -. 
ps ) )
16 nbn2 646 . . . . . 6  |-  ( -. 
ph  ->  ( -.  ps  <->  (
ph 
<->  ps ) ) )
1716adantr 270 . . . . 5  |-  ( ( -.  ph  /\ DECID  ps )  ->  ( -.  ps  <->  ( ph  <->  ps )
) )
1815, 17bitr2d 187 . . . 4  |-  ( ( -.  ph  /\ DECID  ps )  ->  (
( ph  <->  ps )  <->  -.  ( ph 
<->  -.  ps ) ) )
1918ex 113 . . 3  |-  ( -. 
ph  ->  (DECID  ps  ->  ( ( ph 
<->  ps )  <->  -.  ( ph 
<->  -.  ps ) ) ) )
209, 19jaoi 669 . 2  |-  ( (
ph  \/  -.  ph )  ->  (DECID  ps  ->  ( ( ph 
<->  ps )  <->  -.  ( ph 
<->  -.  ps ) ) ) )
211, 20sylbi 119 1  |-  (DECID  ph  ->  (DECID  ps 
->  ( ( ph  <->  ps )  <->  -.  ( ph  <->  -.  ps )
) ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 102    <-> wb 103    \/ wo 662  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:  xor3dc  1319  dfbi3dc  1329
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