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Theorem arg-ax 30887
Description: ? (Contributed by Anthony Hart, 14-Aug-2011.)
Assertion
Ref Expression
arg-ax  |-  ( (
ph  -/\  ( ps  -/\  ch ) )  -/\  (
( ph  -/\  ( ps 
-/\  ch ) )  -/\  ( ( th  -/\  ch )  -/\  ( ( ch  -/\  th )  -/\  ( ph  -/\ 
th ) ) ) ) )

Proof of Theorem arg-ax
StepHypRef Expression
1 df-nan 1380 . . . . 5  |-  ( ( th  -/\  ch )  <->  -.  ( th  /\  ch ) )
2 pm4.57 499 . . . . . . . 8  |-  ( -.  ( -.  ( ch 
/\  th )  /\  -.  ( ph  /\  th )
)  <->  ( ( ch 
/\  th )  \/  ( ph  /\  th ) ) )
3 orel2 384 . . . . . . . . . . . . 13  |-  ( -. 
ph  ->  ( ( ch  \/  ph )  ->  ch ) )
43com12 32 . . . . . . . . . . . 12  |-  ( ( ch  \/  ph )  ->  ( -.  ph  ->  ch ) )
5 simpr 462 . . . . . . . . . . . . 13  |-  ( ( ps  /\  ch )  ->  ch )
65a1i 11 . . . . . . . . . . . 12  |-  ( ( ch  \/  ph )  ->  ( ( ps  /\  ch )  ->  ch )
)
74, 6jad 165 . . . . . . . . . . 11  |-  ( ( ch  \/  ph )  ->  ( ( ph  ->  ( ps  /\  ch )
)  ->  ch )
)
87com12 32 . . . . . . . . . 10  |-  ( (
ph  ->  ( ps  /\  ch ) )  ->  (
( ch  \/  ph )  ->  ch ) )
9 pm3.45 842 . . . . . . . . . . . 12  |-  ( ( ch  ->  ch )  ->  ( ( ch  /\  th )  ->  ( ch  /\ 
th ) ) )
10 pm3.45 842 . . . . . . . . . . . 12  |-  ( (
ph  ->  ch )  -> 
( ( ph  /\  th )  ->  ( ch  /\ 
th ) ) )
119, 10anim12i 568 . . . . . . . . . . 11  |-  ( ( ( ch  ->  ch )  /\  ( ph  ->  ch ) )  ->  (
( ( ch  /\  th )  ->  ( ch  /\ 
th ) )  /\  ( ( ph  /\  th )  ->  ( ch  /\ 
th ) ) ) )
12 jaob 790 . . . . . . . . . . 11  |-  ( ( ( ch  \/  ph )  ->  ch )  <->  ( ( ch  ->  ch )  /\  ( ph  ->  ch )
) )
13 jaob 790 . . . . . . . . . . 11  |-  ( ( ( ( ch  /\  th )  \/  ( ph  /\ 
th ) )  -> 
( ch  /\  th ) )  <->  ( (
( ch  /\  th )  ->  ( ch  /\  th ) )  /\  (
( ph  /\  th )  ->  ( ch  /\  th ) ) ) )
1411, 12, 133imtr4i 269 . . . . . . . . . 10  |-  ( ( ( ch  \/  ph )  ->  ch )  -> 
( ( ( ch 
/\  th )  \/  ( ph  /\  th ) )  ->  ( ch  /\  th ) ) )
158, 14syl 17 . . . . . . . . 9  |-  ( (
ph  ->  ( ps  /\  ch ) )  ->  (
( ( ch  /\  th )  \/  ( ph  /\ 
th ) )  -> 
( ch  /\  th ) ) )
16 pm3.22 450 . . . . . . . . 9  |-  ( ( ch  /\  th )  ->  ( th  /\  ch ) )
1715, 16syl6 34 . . . . . . . 8  |-  ( (
ph  ->  ( ps  /\  ch ) )  ->  (
( ( ch  /\  th )  \/  ( ph  /\ 
th ) )  -> 
( th  /\  ch ) ) )
182, 17syl5bi 220 . . . . . . 7  |-  ( (
ph  ->  ( ps  /\  ch ) )  ->  ( -.  ( -.  ( ch 
/\  th )  /\  -.  ( ph  /\  th )
)  ->  ( th  /\  ch ) ) )
1918con1d 127 . . . . . 6  |-  ( (
ph  ->  ( ps  /\  ch ) )  ->  ( -.  ( th  /\  ch )  ->  ( -.  ( ch  /\  th )  /\  -.  ( ph  /\  th ) ) ) )
20 df-nan 1380 . . . . . . . 8  |-  ( ( ch  -/\  th )  <->  -.  ( ch  /\  th ) )
2120biimpri 209 . . . . . . 7  |-  ( -.  ( ch  /\  th )  ->  ( ch  -/\  th ) )
22 df-nan 1380 . . . . . . . 8  |-  ( (
ph  -/\  th )  <->  -.  ( ph  /\  th ) )
2322biimpri 209 . . . . . . 7  |-  ( -.  ( ph  /\  th )  ->  ( ph  -/\  th )
)
2421, 23anim12i 568 . . . . . 6  |-  ( ( -.  ( ch  /\  th )  /\  -.  ( ph  /\  th ) )  ->  ( ( ch 
-/\  th )  /\  ( ph  -/\  th ) ) )
2519, 24syl6 34 . . . . 5  |-  ( (
ph  ->  ( ps  /\  ch ) )  ->  ( -.  ( th  /\  ch )  ->  ( ( ch 
-/\  th )  /\  ( ph  -/\  th ) ) ) )
261, 25syl5bi 220 . . . 4  |-  ( (
ph  ->  ( ps  /\  ch ) )  ->  (
( th  -/\  ch )  ->  ( ( ch  -/\  th )  /\  ( ph  -/\ 
th ) ) ) )
27 nannan 1384 . . . 4  |-  ( (
ph  -/\  ( ps  -/\  ch ) )  <->  ( ph  ->  ( ps  /\  ch ) ) )
28 nannan 1384 . . . 4  |-  ( ( ( th  -/\  ch )  -/\  ( ( ch  -/\  th )  -/\  ( ph  -/\ 
th ) ) )  <-> 
( ( th  -/\  ch )  ->  ( ( ch  -/\  th )  /\  ( ph  -/\ 
th ) ) ) )
2926, 27, 283imtr4i 269 . . 3  |-  ( (
ph  -/\  ( ps  -/\  ch ) )  ->  (
( th  -/\  ch )  -/\  ( ( ch  -/\  th )  -/\  ( ph  -/\ 
th ) ) ) )
3029ancli 553 . 2  |-  ( (
ph  -/\  ( ps  -/\  ch ) )  ->  (
( ph  -/\  ( ps 
-/\  ch ) )  /\  ( ( th  -/\  ch )  -/\  ( ( ch  -/\  th )  -/\  ( ph  -/\ 
th ) ) ) ) )
31 nannan 1384 . 2  |-  ( ( ( ph  -/\  ( ps  -/\  ch ) ) 
-/\  ( ( ph  -/\  ( ps  -/\  ch )
)  -/\  ( ( th  -/\  ch )  -/\  ( ( ch  -/\  th )  -/\  ( ph  -/\ 
th ) ) ) ) )  <->  ( ( ph  -/\  ( ps  -/\  ch ) )  ->  (
( ph  -/\  ( ps 
-/\  ch ) )  /\  ( ( th  -/\  ch )  -/\  ( ( ch  -/\  th )  -/\  ( ph  -/\ 
th ) ) ) ) ) )
3230, 31mpbir 212 1  |-  ( (
ph  -/\  ( ps  -/\  ch ) )  -/\  (
( ph  -/\  ( ps 
-/\  ch ) )  -/\  ( ( th  -/\  ch )  -/\  ( ( ch  -/\  th )  -/\  ( ph  -/\ 
th ) ) ) ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    \/ 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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