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Theorem rp-fakeanorass 37897
Description: A special case where a mixture of and and or appears to conform to a mixed associative law. (Contributed by Richard Penner, 26-Feb-2020.)
Assertion
Ref Expression
rp-fakeanorass  |-  ( ( ch  ->  ph )  <->  ( (
( ph  /\  ps )  \/  ch )  <->  ( ph  /\  ( ps  \/  ch ) ) ) )

Proof of Theorem rp-fakeanorass
StepHypRef Expression
1 pm1.4 386 . . . . . . . 8  |-  ( (
ph  \/  ch )  ->  ( ch  \/  ph ) )
21ord 377 . . . . . . 7  |-  ( (
ph  \/  ch )  ->  ( -.  ch  ->  ph ) )
3 pm4.83 929 . . . . . . . 8  |-  ( ( ( ch  ->  ph )  /\  ( -.  ch  ->  ph ) )  <->  ph )
43biimpi 194 . . . . . . 7  |-  ( ( ( ch  ->  ph )  /\  ( -.  ch  ->  ph ) )  ->  ph )
52, 4sylan2 474 . . . . . 6  |-  ( ( ( ch  ->  ph )  /\  ( ph  \/  ch ) )  ->  ph )
65ex 434 . . . . 5  |-  ( ( ch  ->  ph )  -> 
( ( ph  \/  ch )  ->  ph )
)
76anim1d 564 . . . 4  |-  ( ( ch  ->  ph )  -> 
( ( ( ph  \/  ch )  /\  ( ps  \/  ch ) )  ->  ( ph  /\  ( ps  \/  ch ) ) ) )
8 orc 385 . . . . 5  |-  ( ph  ->  ( ph  \/  ch ) )
98anim1i 568 . . . 4  |-  ( (
ph  /\  ( ps  \/  ch ) )  -> 
( ( ph  \/  ch )  /\  ( ps  \/  ch ) ) )
107, 9jctir 538 . . 3  |-  ( ( ch  ->  ph )  -> 
( ( ( (
ph  \/  ch )  /\  ( ps  \/  ch ) )  ->  ( ph  /\  ( ps  \/  ch ) ) )  /\  ( ( ph  /\  ( ps  \/  ch ) )  ->  (
( ph  \/  ch )  /\  ( ps  \/  ch ) ) ) ) )
11 olc 384 . . . . . 6  |-  ( ch 
->  ( ph  \/  ch ) )
12 olc 384 . . . . . 6  |-  ( ch 
->  ( ps  \/  ch ) )
1311, 12jca 532 . . . . 5  |-  ( ch 
->  ( ( ph  \/  ch )  /\  ( ps  \/  ch ) ) )
14 simpl 457 . . . . 5  |-  ( (
ph  /\  ( ps  \/  ch ) )  ->  ph )
1513, 14imim12i 57 . . . 4  |-  ( ( ( ( ph  \/  ch )  /\  ( ps  \/  ch ) )  ->  ( ph  /\  ( ps  \/  ch ) ) )  -> 
( ch  ->  ph )
)
1615adantr 465 . . 3  |-  ( ( ( ( ( ph  \/  ch )  /\  ( ps  \/  ch ) )  ->  ( ph  /\  ( ps  \/  ch ) ) )  /\  ( ( ph  /\  ( ps  \/  ch ) )  ->  (
( ph  \/  ch )  /\  ( ps  \/  ch ) ) ) )  ->  ( ch  ->  ph ) )
1710, 16impbii 188 . 2  |-  ( ( ch  ->  ph )  <->  ( (
( ( ph  \/  ch )  /\  ( ps  \/  ch ) )  ->  ( ph  /\  ( ps  \/  ch ) ) )  /\  ( ( ph  /\  ( ps  \/  ch ) )  ->  (
( ph  \/  ch )  /\  ( ps  \/  ch ) ) ) ) )
18 dfbi2 628 . 2  |-  ( ( ( ( ph  \/  ch )  /\  ( ps  \/  ch ) )  <-> 
( ph  /\  ( ps  \/  ch ) ) )  <->  ( ( ( ( ph  \/  ch )  /\  ( ps  \/  ch ) )  ->  ( ph  /\  ( ps  \/  ch ) ) )  /\  ( ( ph  /\  ( ps  \/  ch ) )  ->  (
( ph  \/  ch )  /\  ( ps  \/  ch ) ) ) ) )
19 ordir 865 . . . 4  |-  ( ( ( ph  /\  ps )  \/  ch )  <->  ( ( ph  \/  ch )  /\  ( ps  \/  ch ) ) )
2019bicomi 202 . . 3  |-  ( ( ( ph  \/  ch )  /\  ( ps  \/  ch ) )  <->  ( ( ph  /\  ps )  \/ 
ch ) )
2120bibi1i 314 . 2  |-  ( ( ( ( ph  \/  ch )  /\  ( ps  \/  ch ) )  <-> 
( ph  /\  ( ps  \/  ch ) ) )  <->  ( ( (
ph  /\  ps )  \/  ch )  <->  ( ph  /\  ( ps  \/  ch ) ) ) )
2217, 18, 213bitr2i 273 1  |-  ( ( ch  ->  ph )  <->  ( (
( ph  /\  ps )  \/  ch )  <->  ( ph  /\  ( ps  \/  ch ) ) ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 184    \/ wo 368    /\ wa 369
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 185  df-or 370  df-an 371
This theorem is referenced by:  rp-fakeoranass  37898  rp-fakeinunass  37900
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