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Theorem lukshef-ax2 31074
Description: A single axiom for propositional calculus offered by Lukasiewicz. (Contributed by Anthony Hart, 14-Aug-2011.)
Assertion
Ref Expression
lukshef-ax2  |-  ( (
ph  -/\  ( ps  -/\  ch ) )  -/\  (
( ph  -/\  ( ch 
-/\  ph ) )  -/\  ( ( th  -/\  ps )  -/\  ( ( ph  -/\  th )  -/\  ( ph  -/\  th )
) ) ) )

Proof of Theorem lukshef-ax2
StepHypRef Expression
1 nannan 1385 . . . 4  |-  ( (
ph  -/\  ( ps  -/\  ch ) )  <->  ( ph  ->  ( ps  /\  ch ) ) )
21biimpi 198 . . 3  |-  ( (
ph  -/\  ( ps  -/\  ch ) )  ->  ( ph  ->  ( ps  /\  ch ) ) )
3 simpr 463 . . . . 5  |-  ( ( ps  /\  ch )  ->  ch )
43imim2i 16 . . . 4  |-  ( (
ph  ->  ( ps  /\  ch ) )  ->  ( ph  ->  ch ) )
5 simpl 459 . . . . . 6  |-  ( ( ps  /\  ch )  ->  ps )
65imim2i 16 . . . . 5  |-  ( (
ph  ->  ( ps  /\  ch ) )  ->  ( ph  ->  ps ) )
7 pm2.27 41 . . . . . . 7  |-  ( ph  ->  ( ( ph  ->  ps )  ->  ps )
)
87anim2d 568 . . . . . 6  |-  ( ph  ->  ( ( th  /\  ( ph  ->  ps )
)  ->  ( th  /\  ps ) ) )
98expdimp 439 . . . . 5  |-  ( (
ph  /\  th )  ->  ( ( ph  ->  ps )  ->  ( th  /\  ps ) ) )
106, 9syl5com 32 . . . 4  |-  ( (
ph  ->  ( ps  /\  ch ) )  ->  (
( ph  /\  th )  ->  ( th  /\  ps ) ) )
11 ancr 552 . . . . 5  |-  ( (
ph  ->  ch )  -> 
( ph  ->  ( ch 
/\  ph ) ) )
1211anim1i 571 . . . 4  |-  ( ( ( ph  ->  ch )  /\  ( ( ph  /\ 
th )  ->  ( th  /\  ps ) ) )  ->  ( ( ph  ->  ( ch  /\  ph ) )  /\  (
( ph  /\  th )  ->  ( th  /\  ps ) ) ) )
134, 10, 12syl2anc 666 . . 3  |-  ( (
ph  ->  ( ps  /\  ch ) )  ->  (
( ph  ->  ( ch 
/\  ph ) )  /\  ( ( ph  /\  th )  ->  ( th  /\  ps ) ) ) )
14 con3 140 . . . . 5  |-  ( ( ( ph  /\  th )  ->  ( th  /\  ps ) )  ->  ( -.  ( th  /\  ps )  ->  -.  ( ph  /\ 
th ) ) )
15 df-nan 1381 . . . . 5  |-  ( ( th  -/\  ps )  <->  -.  ( th  /\  ps ) )
16 df-nan 1381 . . . . 5  |-  ( (
ph  -/\  th )  <->  -.  ( ph  /\  th ) )
1714, 15, 163imtr4g 274 . . . 4  |-  ( ( ( ph  /\  th )  ->  ( th  /\  ps ) )  ->  (
( th  -/\  ps )  ->  ( ph  -/\  th )
) )
1817anim2i 572 . . 3  |-  ( ( ( ph  ->  ( ch  /\  ph ) )  /\  ( ( ph  /\ 
th )  ->  ( th  /\  ps ) ) )  ->  ( ( ph  ->  ( ch  /\  ph ) )  /\  (
( th  -/\  ps )  ->  ( ph  -/\  th )
) ) )
19 nannan 1385 . . . . 5  |-  ( (
ph  -/\  ( ch  -/\  ph ) )  <->  ( ph  ->  ( ch  /\  ph ) ) )
2019biimpri 210 . . . 4  |-  ( (
ph  ->  ( ch  /\  ph ) )  ->  ( ph  -/\  ( ch  -/\  ph ) ) )
21 nanim 1387 . . . . 5  |-  ( ( ( th  -/\  ps )  ->  ( ph  -/\  th )
)  <->  ( ( th 
-/\  ps )  -/\  (
( ph  -/\  th )  -/\  ( ph  -/\  th )
) ) )
2221biimpi 198 . . . 4  |-  ( ( ( th  -/\  ps )  ->  ( ph  -/\  th )
)  ->  ( ( th  -/\  ps )  -/\  ( ( ph  -/\  th )  -/\  ( ph  -/\  th )
) ) )
2320, 22anim12i 569 . . 3  |-  ( ( ( ph  ->  ( ch  /\  ph ) )  /\  ( ( th 
-/\  ps )  ->  ( ph  -/\  th ) ) )  ->  ( ( ph  -/\  ( ch  -/\  ph ) )  /\  (
( th  -/\  ps )  -/\  ( ( ph  -/\  th )  -/\  ( ph  -/\  th )
) ) ) )
242, 13, 18, 234syl 19 . 2  |-  ( (
ph  -/\  ( ps  -/\  ch ) )  ->  (
( ph  -/\  ( ch 
-/\  ph ) )  /\  ( ( th  -/\  ps )  -/\  ( ( ph  -/\  th )  -/\  ( ph  -/\  th )
) ) ) )
25 nannan 1385 . 2  |-  ( ( ( ph  -/\  ( ps  -/\  ch ) ) 
-/\  ( ( ph  -/\  ( ch  -/\  ph )
)  -/\  ( ( th  -/\  ps )  -/\  ( ( ph  -/\  th )  -/\  ( ph  -/\  th )
) ) ) )  <-> 
( ( ph  -/\  ( ps  -/\  ch ) )  ->  ( ( ph  -/\  ( ch  -/\  ph )
)  /\  ( ( th  -/\  ps )  -/\  ( ( ph  -/\  th )  -/\  ( ph  -/\  th )
) ) ) ) )
2624, 25mpbir 213 1  |-  ( (
ph  -/\  ( ps  -/\  ch ) )  -/\  (
( ph  -/\  ( ch 
-/\  ph ) )  -/\  ( ( th  -/\  ps )  -/\  ( ( ph  -/\  th )  -/\  ( ph  -/\  th )
) ) ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 371    -/\ wnan 1380
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-an 373  df-nan 1381
This theorem is referenced by: (None)
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