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Theorem waj-ax 31018
Description: A single axiom for propositional calculus offered by Wajsberg. (Contributed by Anthony Hart, 13-Aug-2011.)
Assertion
Ref Expression
waj-ax  |-  ( (
ph  -/\  ( ps  -/\  ch ) )  -/\  (
( ( th  -/\  ch )  -/\  ( ( ph  -/\  th )  -/\  ( ph  -/\  th )
) )  -/\  ( ph  -/\  ( ph  -/\  ps )
) ) )

Proof of Theorem waj-ax
StepHypRef Expression
1 nannan 1384 . . 3  |-  ( (
ph  -/\  ( ps  -/\  ch ) )  <->  ( ph  ->  ( ps  /\  ch ) ) )
2 simpr 462 . . . . . . . . 9  |-  ( ( ps  /\  ch )  ->  ch )
32imim2i 16 . . . . . . . 8  |-  ( (
ph  ->  ( ps  /\  ch ) )  ->  ( ph  ->  ch ) )
4 pm2.27 40 . . . . . . . . . 10  |-  ( ph  ->  ( ( ph  ->  ch )  ->  ch )
)
54anim2d 567 . . . . . . . . 9  |-  ( ph  ->  ( ( th  /\  ( ph  ->  ch )
)  ->  ( th  /\  ch ) ) )
65expdimp 438 . . . . . . . 8  |-  ( (
ph  /\  th )  ->  ( ( ph  ->  ch )  ->  ( th  /\  ch ) ) )
73, 6syl5com 31 . . . . . . 7  |-  ( (
ph  ->  ( ps  /\  ch ) )  ->  (
( ph  /\  th )  ->  ( th  /\  ch ) ) )
87con3d 138 . . . . . 6  |-  ( (
ph  ->  ( ps  /\  ch ) )  ->  ( -.  ( th  /\  ch )  ->  -.  ( ph  /\ 
th ) ) )
9 df-nan 1380 . . . . . 6  |-  ( ( th  -/\  ch )  <->  -.  ( th  /\  ch ) )
10 df-nan 1380 . . . . . 6  |-  ( (
ph  -/\  th )  <->  -.  ( ph  /\  th ) )
118, 9, 103imtr4g 273 . . . . 5  |-  ( (
ph  ->  ( ps  /\  ch ) )  ->  (
( th  -/\  ch )  ->  ( ph  -/\  th )
) )
12 nanim 1386 . . . . 5  |-  ( ( ( th  -/\  ch )  ->  ( ph  -/\  th )
)  <->  ( ( th 
-/\  ch )  -/\  (
( ph  -/\  th )  -/\  ( ph  -/\  th )
) ) )
1311, 12sylib 199 . . . 4  |-  ( (
ph  ->  ( ps  /\  ch ) )  ->  (
( th  -/\  ch )  -/\  ( ( ph  -/\  th )  -/\  ( ph  -/\  th )
) ) )
14 pm3.21 449 . . . . . . . 8  |-  ( ps 
->  ( ph  ->  ( ph  /\  ps ) ) )
1514adantr 466 . . . . . . 7  |-  ( ( ps  /\  ch )  ->  ( ph  ->  ( ph  /\  ps ) ) )
1615com12 32 . . . . . 6  |-  ( ph  ->  ( ( ps  /\  ch )  ->  ( ph  /\ 
ps ) ) )
1716a2i 14 . . . . 5  |-  ( (
ph  ->  ( ps  /\  ch ) )  ->  ( ph  ->  ( ph  /\  ps ) ) )
18 nannan 1384 . . . . 5  |-  ( (
ph  -/\  ( ph  -/\  ps )
)  <->  ( ph  ->  (
ph  /\  ps )
) )
1917, 18sylibr 215 . . . 4  |-  ( (
ph  ->  ( ps  /\  ch ) )  ->  ( ph  -/\  ( ph  -/\  ps )
) )
2013, 19jca 534 . . 3  |-  ( (
ph  ->  ( ps  /\  ch ) )  ->  (
( ( th  -/\  ch )  -/\  ( ( ph  -/\  th )  -/\  ( ph  -/\  th )
) )  /\  ( ph  -/\  ( ph  -/\  ps )
) ) )
211, 20sylbi 198 . 2  |-  ( (
ph  -/\  ( ps  -/\  ch ) )  ->  (
( ( th  -/\  ch )  -/\  ( ( ph  -/\  th )  -/\  ( ph  -/\  th )
) )  /\  ( ph  -/\  ( ph  -/\  ps )
) ) )
22 nannan 1384 . 2  |-  ( ( ( ph  -/\  ( ps  -/\  ch ) ) 
-/\  ( ( ( th  -/\  ch )  -/\  ( ( ph  -/\  th )  -/\  ( ph  -/\  th )
) )  -/\  ( ph  -/\  ( ph  -/\  ps )
) ) )  <->  ( ( ph  -/\  ( ps  -/\  ch ) )  ->  (
( ( th  -/\  ch )  -/\  ( ( ph  -/\  th )  -/\  ( ph  -/\  th )
) )  /\  ( ph  -/\  ( ph  -/\  ps )
) ) ) )
2321, 22mpbir 212 1  |-  ( (
ph  -/\  ( ps  -/\  ch ) )  -/\  (
( ( th  -/\  ch )  -/\  ( ( ph  -/\  th )  -/\  ( ph  -/\  th )
) )  -/\  ( ph  -/\  ( ph  -/\  ps )
) ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    /\ 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-an 372  df-nan 1380
This theorem is referenced by: (None)
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