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Theorem rp-fakeimass 36168
Description: A special case where implication appears to conform to a mixed associative law. (Contributed by Richard Penner, 29-Feb-2020.)
Assertion
Ref Expression
rp-fakeimass  |-  ( (
ph  \/  ch )  <->  ( ( ( ph  ->  ps )  ->  ch )  <->  (
ph  ->  ( ps  ->  ch ) ) ) )

Proof of Theorem rp-fakeimass
StepHypRef Expression
1 ax-1 6 . . . . . . . 8  |-  ( ps 
->  ( ph  ->  ps ) )
21con3i 141 . . . . . . 7  |-  ( -.  ( ph  ->  ps )  ->  -.  ps )
32pm2.21d 110 . . . . . 6  |-  ( -.  ( ph  ->  ps )  ->  ( ps  ->  ch ) )
43a1d 26 . . . . 5  |-  ( -.  ( ph  ->  ps )  ->  ( ph  ->  ( ps  ->  ch )
) )
5 ax-1 6 . . . . . 6  |-  ( ch 
->  ( ps  ->  ch ) )
65a1d 26 . . . . 5  |-  ( ch 
->  ( ph  ->  ( ps  ->  ch ) ) )
74, 6ja 165 . . . 4  |-  ( ( ( ph  ->  ps )  ->  ch )  -> 
( ph  ->  ( ps 
->  ch ) ) )
8 ax-2 7 . . . . 5  |-  ( (
ph  ->  ( ps  ->  ch ) )  ->  (
( ph  ->  ps )  ->  ( ph  ->  ch ) ) )
98com3r 82 . . . 4  |-  ( ph  ->  ( ( ph  ->  ( ps  ->  ch )
)  ->  ( ( ph  ->  ps )  ->  ch ) ) )
107, 9impbid2 208 . . 3  |-  ( ph  ->  ( ( ( ph  ->  ps )  ->  ch ) 
<->  ( ph  ->  ( ps  ->  ch ) ) ) )
11 ax-1 6 . . . 4  |-  ( ch 
->  ( ( ph  ->  ps )  ->  ch )
)
1211, 62thd 244 . . 3  |-  ( ch 
->  ( ( ( ph  ->  ps )  ->  ch ) 
<->  ( ph  ->  ( ps  ->  ch ) ) ) )
1310, 12jaoi 381 . 2  |-  ( (
ph  \/  ch )  ->  ( ( ( ph  ->  ps )  ->  ch ) 
<->  ( ph  ->  ( ps  ->  ch ) ) ) )
14 jarl 167 . . . . 5  |-  ( ( ( ph  ->  ps )  ->  ch )  -> 
( -.  ph  ->  ch ) )
1514orrd 380 . . . 4  |-  ( ( ( ph  ->  ps )  ->  ch )  -> 
( ph  \/  ch ) )
1615a1d 26 . . 3  |-  ( ( ( ph  ->  ps )  ->  ch )  -> 
( ( ph  ->  ( ps  ->  ch )
)  ->  ( ph  \/  ch ) ) )
17 simplim 155 . . . . 5  |-  ( -.  ( ph  ->  ( ps  ->  ch ) )  ->  ph )
1817orcd 394 . . . 4  |-  ( -.  ( ph  ->  ( ps  ->  ch ) )  ->  ( ph  \/  ch ) )
1918a1i 11 . . 3  |-  ( -.  ( ( ph  ->  ps )  ->  ch )  ->  ( -.  ( ph  ->  ( ps  ->  ch ) )  ->  ( ph  \/  ch ) ) )
2016, 19bija 357 . 2  |-  ( ( ( ( ph  ->  ps )  ->  ch )  <->  (
ph  ->  ( ps  ->  ch ) ) )  -> 
( ph  \/  ch ) )
2113, 20impbii 191 1  |-  ( (
ph  \/  ch )  <->  ( ( ( ph  ->  ps )  ->  ch )  <->  (
ph  ->  ( ps  ->  ch ) ) ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 188    \/ wo 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 189  df-or 372
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator