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Theorem lukshef-ax2 24854
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 1291 . . . 4  |-  ( (
ph  -/\  ( ps  -/\  ch ) )  <->  ( ph  ->  ( ps  /\  ch ) ) )
21biimpi 186 . . 3  |-  ( (
ph  -/\  ( ps  -/\  ch ) )  ->  ( ph  ->  ( ps  /\  ch ) ) )
3 simpr 447 . . . . . 6  |-  ( ( ps  /\  ch )  ->  ch )
43imim2i 13 . . . . 5  |-  ( (
ph  ->  ( ps  /\  ch ) )  ->  ( ph  ->  ch ) )
5 simpl 443 . . . . . . 7  |-  ( ( ps  /\  ch )  ->  ps )
65imim2i 13 . . . . . 6  |-  ( (
ph  ->  ( ps  /\  ch ) )  ->  ( ph  ->  ps ) )
7 pm2.27 35 . . . . . . . 8  |-  ( ph  ->  ( ( ph  ->  ps )  ->  ps )
)
87anim2d 548 . . . . . . 7  |-  ( ph  ->  ( ( th  /\  ( ph  ->  ps )
)  ->  ( th  /\  ps ) ) )
98expdimp 426 . . . . . 6  |-  ( (
ph  /\  th )  ->  ( ( ph  ->  ps )  ->  ( th  /\  ps ) ) )
106, 9syl5com 26 . . . . 5  |-  ( (
ph  ->  ( ps  /\  ch ) )  ->  (
( ph  /\  th )  ->  ( th  /\  ps ) ) )
11 ancr 532 . . . . . 6  |-  ( (
ph  ->  ch )  -> 
( ph  ->  ( ch 
/\  ph ) ) )
1211anim1i 551 . . . . 5  |-  ( ( ( ph  ->  ch )  /\  ( ( ph  /\ 
th )  ->  ( th  /\  ps ) ) )  ->  ( ( ph  ->  ( ch  /\  ph ) )  /\  (
( ph  /\  th )  ->  ( th  /\  ps ) ) ) )
134, 10, 12syl2anc 642 . . . 4  |-  ( (
ph  ->  ( ps  /\  ch ) )  ->  (
( ph  ->  ( ch 
/\  ph ) )  /\  ( ( ph  /\  th )  ->  ( th  /\  ps ) ) ) )
14 con3 126 . . . . . 6  |-  ( ( ( ph  /\  th )  ->  ( th  /\  ps ) )  ->  ( -.  ( th  /\  ps )  ->  -.  ( ph  /\ 
th ) ) )
15 df-nan 1288 . . . . . 6  |-  ( ( th  -/\  ps )  <->  -.  ( th  /\  ps ) )
16 df-nan 1288 . . . . . 6  |-  ( (
ph  -/\  th )  <->  -.  ( ph  /\  th ) )
1714, 15, 163imtr4g 261 . . . . 5  |-  ( ( ( ph  /\  th )  ->  ( th  /\  ps ) )  ->  (
( th  -/\  ps )  ->  ( ph  -/\  th )
) )
1817anim2i 552 . . . 4  |-  ( ( ( ph  ->  ( ch  /\  ph ) )  /\  ( ( ph  /\ 
th )  ->  ( th  /\  ps ) ) )  ->  ( ( ph  ->  ( ch  /\  ph ) )  /\  (
( th  -/\  ps )  ->  ( ph  -/\  th )
) ) )
1913, 18syl 15 . . 3  |-  ( (
ph  ->  ( ps  /\  ch ) )  ->  (
( ph  ->  ( ch 
/\  ph ) )  /\  ( ( th  -/\  ps )  ->  ( ph  -/\  th )
) ) )
20 nannan 1291 . . . . 5  |-  ( (
ph  -/\  ( ch  -/\  ph ) )  <->  ( ph  ->  ( ch  /\  ph ) ) )
2120biimpri 197 . . . 4  |-  ( (
ph  ->  ( ch  /\  ph ) )  ->  ( ph  -/\  ( ch  -/\  ph ) ) )
22 nanim 1292 . . . . 5  |-  ( ( ( th  -/\  ps )  ->  ( ph  -/\  th )
)  <->  ( ( th 
-/\  ps )  -/\  (
( ph  -/\  th )  -/\  ( ph  -/\  th )
) ) )
2322biimpi 186 . . . 4  |-  ( ( ( th  -/\  ps )  ->  ( ph  -/\  th )
)  ->  ( ( th  -/\  ps )  -/\  ( ( ph  -/\  th )  -/\  ( ph  -/\  th )
) ) )
2421, 23anim12i 549 . . 3  |-  ( ( ( ph  ->  ( ch  /\  ph ) )  /\  ( ( th 
-/\  ps )  ->  ( ph  -/\  th ) ) )  ->  ( ( ph  -/\  ( ch  -/\  ph ) )  /\  (
( th  -/\  ps )  -/\  ( ( ph  -/\  th )  -/\  ( ph  -/\  th )
) ) ) )
252, 19, 243syl 18 . 2  |-  ( (
ph  -/\  ( ps  -/\  ch ) )  ->  (
( ph  -/\  ( ch 
-/\  ph ) )  /\  ( ( th  -/\  ps )  -/\  ( ( ph  -/\  th )  -/\  ( ph  -/\  th )
) ) ) )
26 nannan 1291 . 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 )
) ) ) ) )
2725, 26mpbir 200 1  |-  ( (
ph  -/\  ( ps  -/\  ch ) )  -/\  (
( ph  -/\  ( ch 
-/\  ph ) )  -/\  ( ( th  -/\  ps )  -/\  ( ( ph  -/\  th )  -/\  ( ph  -/\  th )
) ) ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 358    -/\ wnan 1287
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8
This theorem depends on definitions:  df-bi 177  df-an 360  df-nan 1288
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