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Theorem rp-fakenanass 36129
Description: A special case where nand appears to conform to a mixed associative law. (Contributed by Richard Penner, 29-Feb-2020.)
Assertion
Ref Expression
rp-fakenanass  |-  ( (
ph 
<->  ch )  <->  ( (
( ph  -/\  ps )  -/\  ch )  <->  ( ph  -/\  ( ps  -/\  ch )
) ) )

Proof of Theorem rp-fakenanass
StepHypRef Expression
1 bicom1 202 . . . 4  |-  ( (
ph 
<->  ch )  ->  ( ch 
<-> 
ph ) )
2 nanbi2 1392 . . . 4  |-  ( (
ph 
<->  ch )  ->  (
( ps  -/\  ph )  <->  ( ps  -/\  ch )
) )
31, 2nanbi12d 1399 . . 3  |-  ( (
ph 
<->  ch )  ->  (
( ch  -/\  ( ps  -/\  ph ) )  <->  ( ph  -/\  ( ps  -/\  ch )
) ) )
4 nannan 1384 . . . . . 6  |-  ( (
ph  -/\  ( ps  -/\  ch ) )  <->  ( ph  ->  ( ps  /\  ch ) ) )
5 simpr 462 . . . . . . 7  |-  ( ( ps  /\  ch )  ->  ch )
65imim2i 16 . . . . . 6  |-  ( (
ph  ->  ( ps  /\  ch ) )  ->  ( ph  ->  ch ) )
74, 6sylbi 198 . . . . 5  |-  ( (
ph  -/\  ( ps  -/\  ch ) )  ->  ( ph  ->  ch ) )
8 nannan 1384 . . . . . 6  |-  ( ( ch  -/\  ( ps  -/\  ph ) )  <->  ( ch  ->  ( ps  /\  ph ) ) )
9 simpr 462 . . . . . . 7  |-  ( ( ps  /\  ph )  ->  ph )
109imim2i 16 . . . . . 6  |-  ( ( ch  ->  ( ps  /\ 
ph ) )  -> 
( ch  ->  ph )
)
118, 10sylbi 198 . . . . 5  |-  ( ( ch  -/\  ( ps  -/\  ph ) )  ->  ( ch  ->  ph ) )
127, 11impbid21d 192 . . . 4  |-  ( ( ch  -/\  ( ps  -/\  ph ) )  ->  (
( ph  -/\  ( ps 
-/\  ch ) )  -> 
( ph  <->  ch ) ) )
138notbii 297 . . . . . . 7  |-  ( -.  ( ch  -/\  ( ps  -/\  ph ) )  <->  -.  ( ch  ->  ( ps  /\  ph ) ) )
14 pm4.61 427 . . . . . . 7  |-  ( -.  ( ch  ->  ( ps  /\  ph ) )  <-> 
( ch  /\  -.  ( ps  /\  ph )
) )
15 ianor 490 . . . . . . . 8  |-  ( -.  ( ps  /\  ph ) 
<->  ( -.  ps  \/  -.  ph ) )
1615anbi2i 698 . . . . . . 7  |-  ( ( ch  /\  -.  ( ps  /\  ph ) )  <-> 
( ch  /\  ( -.  ps  \/  -.  ph ) ) )
1713, 14, 163bitri 274 . . . . . 6  |-  ( -.  ( ch  -/\  ( ps  -/\  ph ) )  <->  ( ch  /\  ( -.  ps  \/  -.  ph ) ) )
184notbii 297 . . . . . . 7  |-  ( -.  ( ph  -/\  ( ps  -/\  ch ) )  <->  -.  ( ph  ->  ( ps  /\  ch ) ) )
19 pm4.61 427 . . . . . . 7  |-  ( -.  ( ph  ->  ( ps  /\  ch ) )  <-> 
( ph  /\  -.  ( ps  /\  ch ) ) )
20 ianor 490 . . . . . . . 8  |-  ( -.  ( ps  /\  ch ) 
<->  ( -.  ps  \/  -.  ch ) )
2120anbi2i 698 . . . . . . 7  |-  ( (
ph  /\  -.  ( ps  /\  ch ) )  <-> 
( ph  /\  ( -.  ps  \/  -.  ch ) ) )
2218, 19, 213bitri 274 . . . . . 6  |-  ( -.  ( ph  -/\  ( ps  -/\  ch ) )  <-> 
( ph  /\  ( -.  ps  \/  -.  ch ) ) )
23 pm5.1 865 . . . . . . . 8  |-  ( (
ph  /\  ch )  ->  ( ph  <->  ch )
)
2423ancoms 454 . . . . . . 7  |-  ( ( ch  /\  ph )  ->  ( ph  <->  ch )
)
2524ad2ant2r 751 . . . . . 6  |-  ( ( ( ch  /\  ( -.  ps  \/  -.  ph ) )  /\  ( ph  /\  ( -.  ps  \/  -.  ch ) ) )  ->  ( ph  <->  ch ) )
2617, 22, 25syl2anb 481 . . . . 5  |-  ( ( -.  ( ch  -/\  ( ps  -/\  ph )
)  /\  -.  ( ph  -/\  ( ps  -/\  ch ) ) )  -> 
( ph  <->  ch ) )
2726ex 435 . . . 4  |-  ( -.  ( ch  -/\  ( ps  -/\  ph ) )  -> 
( -.  ( ph  -/\  ( ps  -/\  ch )
)  ->  ( ph  <->  ch ) ) )
2812, 27bija 356 . . 3  |-  ( ( ( ch  -/\  ( ps  -/\  ph ) )  <->  ( ph  -/\  ( ps  -/\  ch )
) )  ->  ( ph 
<->  ch ) )
293, 28impbii 190 . 2  |-  ( (
ph 
<->  ch )  <->  ( ( ch  -/\  ( ps  -/\  ph ) )  <->  ( ph  -/\  ( ps  -/\  ch )
) ) )
30 nancom 1382 . . . . 5  |-  ( ( ps  -/\  ph )  <->  ( ph  -/\ 
ps ) )
3130nanbi2i 1395 . . . 4  |-  ( ( ch  -/\  ( ps  -/\  ph ) )  <->  ( ch  -/\  ( ph  -/\  ps )
) )
32 nancom 1382 . . . 4  |-  ( ( ch  -/\  ( ph  -/\ 
ps ) )  <->  ( ( ph  -/\  ps )  -/\  ch ) )
3331, 32bitri 252 . . 3  |-  ( ( ch  -/\  ( ps  -/\  ph ) )  <->  ( ( ph  -/\  ps )  -/\  ch ) )
3433bibi1i 315 . 2  |-  ( ( ( ch  -/\  ( ps  -/\  ph ) )  <->  ( ph  -/\  ( ps  -/\  ch )
) )  <->  ( (
( ph  -/\  ps )  -/\  ch )  <->  ( ph  -/\  ( ps  -/\  ch )
) ) )
3529, 34bitri 252 1  |-  ( (
ph 
<->  ch )  <->  ( (
( ph  -/\  ps )  -/\  ch )  <->  ( ph  -/\  ( ps  -/\  ch )
) ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 187    \/ wo 369    /\ wa 370    -/\ wnan 1379
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-or 371  df-an 372  df-nan 1380
This theorem is referenced by: (None)
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