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Theorem confun5 38402
Description: An attempt at derivative. Resisted simplest path to a proof. Interesting that ch, th, ta, et were all provable. (Contributed by Jarvin Udandy, 7-Sep-2020.)
Hypotheses
Ref Expression
confun5.1  |-  ph
confun5.2  |-  ( (
ph  ->  ps )  ->  ps )
confun5.3  |-  ( ps 
->  ( ph  ->  ch ) )
confun5.4  |-  ( ( ch  ->  th )  ->  ( ( ph  ->  th )  <->  ps ) )
confun5.5  |-  ( ta  <->  ( ch  ->  th )
)
confun5.6  |-  ( et  <->  -.  ( ch  ->  ( ch  /\  -.  ch )
) )
confun5.7  |-  ps
confun5.8  |-  ( ch 
->  th )
Assertion
Ref Expression
confun5  |-  ( ch 
->  ( et  <->  ta )
)

Proof of Theorem confun5
StepHypRef Expression
1 confun5.1 . . . . . 6  |-  ph
2 confun5.7 . . . . . . 7  |-  ps
3 confun5.3 . . . . . . 7  |-  ( ps 
->  ( ph  ->  ch ) )
42, 3ax-mp 5 . . . . . 6  |-  ( ph  ->  ch )
51, 4ax-mp 5 . . . . 5  |-  ch
65atnaiana 38382 . . . 4  |-  -.  ( ch  ->  ( ch  /\  -.  ch ) )
7 confun5.6 . . . . . 6  |-  ( et  <->  -.  ( ch  ->  ( ch  /\  -.  ch )
) )
8 bicom1 202 . . . . . 6  |-  ( ( et  <->  -.  ( ch  ->  ( ch  /\  -.  ch ) ) )  -> 
( -.  ( ch 
->  ( ch  /\  -.  ch ) )  <->  et )
)
97, 8ax-mp 5 . . . . 5  |-  ( -.  ( ch  ->  ( ch  /\  -.  ch )
)  <->  et )
109biimpi 197 . . . 4  |-  ( -.  ( ch  ->  ( ch  /\  -.  ch )
)  ->  et )
116, 10ax-mp 5 . . 3  |-  et
12 confun5.8 . . . 4  |-  ( ch 
->  th )
13 confun5.5 . . . . . 6  |-  ( ta  <->  ( ch  ->  th )
)
14 bicom1 202 . . . . . 6  |-  ( ( ta  <->  ( ch  ->  th ) )  ->  (
( ch  ->  th )  <->  ta ) )
1513, 14ax-mp 5 . . . . 5  |-  ( ( ch  ->  th )  <->  ta )
1615biimpi 197 . . . 4  |-  ( ( ch  ->  th )  ->  ta )
1712, 16ax-mp 5 . . 3  |-  ta
1811, 172th 242 . 2  |-  ( et  <->  ta )
19 ax-1 6 . 2  |-  ( ( et  <->  ta )  ->  ( ch  ->  ( et  <->  ta )
) )
2018, 19ax-mp 5 1  |-  ( ch 
->  ( et  <->  ta )
)
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 187    /\ wa 370
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8
This theorem depends on definitions:  df-bi 188  df-an 372  df-tru 1440  df-fal 1443
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator