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Theorem nic-axALT 1429
Description: A direct proof of nic-ax 1428. (Contributed by NM, 11-Dec-2008.) (Proof modification is discouraged.) (New usage is discouraged.)
Assertion
Ref Expression
nic-axALT  |-  ( (
ph  -/\  ( ch  -/\  ps ) )  -/\  (
( ta  -/\  ( ta  -/\  ta ) ) 
-/\  ( ( th 
-/\  ch )  -/\  (
( ph  -/\  th )  -/\  ( ph  -/\  th )
) ) ) )

Proof of Theorem nic-axALT
StepHypRef Expression
1 simpl 443 . . . . . 6  |-  ( ( ch  /\  ps )  ->  ch )
21imim2i 13 . . . . 5  |-  ( (
ph  ->  ( ch  /\  ps ) )  ->  ( ph  ->  ch ) )
3 con3 126 . . . . . 6  |-  ( (
ph  ->  ch )  -> 
( -.  ch  ->  -. 
ph ) )
43imim2d 48 . . . . 5  |-  ( (
ph  ->  ch )  -> 
( ( th  ->  -. 
ch )  ->  ( th  ->  -.  ph ) ) )
52, 4syl 15 . . . 4  |-  ( (
ph  ->  ( ch  /\  ps ) )  ->  (
( th  ->  -.  ch )  ->  ( th 
->  -.  ph ) ) )
6 anidm 625 . . . . 5  |-  ( ( ta  /\  ta )  <->  ta )
76biimpri 197 . . . 4  |-  ( ta 
->  ( ta  /\  ta ) )
85, 7jctil 523 . . 3  |-  ( (
ph  ->  ( ch  /\  ps ) )  ->  (
( ta  ->  ( ta  /\  ta ) )  /\  ( ( th 
->  -.  ch )  -> 
( th  ->  -.  ph ) ) ) )
9 df-nan 1288 . . . . . . . . 9  |-  ( ( ch  -/\  ps )  <->  -.  ( ch  /\  ps ) )
109anbi2i 675 . . . . . . . 8  |-  ( (
ph  /\  ( ch  -/\ 
ps ) )  <->  ( ph  /\ 
-.  ( ch  /\  ps ) ) )
1110notbii 287 . . . . . . 7  |-  ( -.  ( ph  /\  ( ch  -/\  ps ) )  <->  -.  ( ph  /\  -.  ( ch  /\  ps )
) )
12 df-nan 1288 . . . . . . 7  |-  ( (
ph  -/\  ( ch  -/\  ps ) )  <->  -.  ( ph  /\  ( ch  -/\  ps ) ) )
13 iman 413 . . . . . . 7  |-  ( (
ph  ->  ( ch  /\  ps ) )  <->  -.  ( ph  /\  -.  ( ch 
/\  ps ) ) )
1411, 12, 133bitr4i 268 . . . . . 6  |-  ( (
ph  -/\  ( ch  -/\  ps ) )  <->  ( ph  ->  ( ch  /\  ps ) ) )
15 df-nan 1288 . . . . . . 7  |-  ( ( ( ta  -/\  ( ta  -/\  ta ) ) 
-/\  ( ( th 
-/\  ch )  -/\  (
( ph  -/\  th )  -/\  ( ph  -/\  th )
) ) )  <->  -.  (
( ta  -/\  ( ta  -/\  ta ) )  /\  ( ( th 
-/\  ch )  -/\  (
( ph  -/\  th )  -/\  ( ph  -/\  th )
) ) ) )
16 df-nan 1288 . . . . . . . . . . 11  |-  ( ( ta  -/\  ta )  <->  -.  ( ta  /\  ta ) )
1716anbi2i 675 . . . . . . . . . 10  |-  ( ( ta  /\  ( ta 
-/\  ta ) )  <->  ( ta  /\ 
-.  ( ta  /\  ta ) ) )
1817notbii 287 . . . . . . . . 9  |-  ( -.  ( ta  /\  ( ta  -/\  ta ) )  <->  -.  ( ta  /\  -.  ( ta  /\  ta )
) )
19 df-nan 1288 . . . . . . . . 9  |-  ( ( ta  -/\  ( ta  -/\ 
ta ) )  <->  -.  ( ta  /\  ( ta  -/\  ta ) ) )
20 iman 413 . . . . . . . . 9  |-  ( ( ta  ->  ( ta  /\ 
ta ) )  <->  -.  ( ta  /\  -.  ( ta 
/\  ta ) ) )
2118, 19, 203bitr4i 268 . . . . . . . 8  |-  ( ( ta  -/\  ( ta  -/\ 
ta ) )  <->  ( ta  ->  ( ta  /\  ta ) ) )
22 df-nan 1288 . . . . . . . . . . . 12  |-  ( ( th  -/\  ch )  <->  -.  ( th  /\  ch ) )
23 imnan 411 . . . . . . . . . . . 12  |-  ( ( th  ->  -.  ch )  <->  -.  ( th  /\  ch ) )
2422, 23bitr4i 243 . . . . . . . . . . 11  |-  ( ( th  -/\  ch )  <->  ( th  ->  -.  ch )
)
25 df-nan 1288 . . . . . . . . . . . 12  |-  ( ( ( ph  -/\  th )  -/\  ( ph  -/\  th )
)  <->  -.  ( ( ph  -/\  th )  /\  ( ph  -/\  th )
) )
26 anidm 625 . . . . . . . . . . . . 13  |-  ( ( ( ph  -/\  th )  /\  ( ph  -/\  th )
)  <->  ( ph  -/\  th )
)
27 df-nan 1288 . . . . . . . . . . . . 13  |-  ( (
ph  -/\  th )  <->  -.  ( ph  /\  th ) )
28 imnan 411 . . . . . . . . . . . . . 14  |-  ( (
ph  ->  -.  th )  <->  -.  ( ph  /\  th ) )
29 con2b 324 . . . . . . . . . . . . . 14  |-  ( (
ph  ->  -.  th )  <->  ( th  ->  -.  ph )
)
3028, 29bitr3i 242 . . . . . . . . . . . . 13  |-  ( -.  ( ph  /\  th ) 
<->  ( th  ->  -.  ph ) )
3126, 27, 303bitri 262 . . . . . . . . . . . 12  |-  ( ( ( ph  -/\  th )  /\  ( ph  -/\  th )
)  <->  ( th  ->  -. 
ph ) )
3225, 31xchbinx 301 . . . . . . . . . . 11  |-  ( ( ( ph  -/\  th )  -/\  ( ph  -/\  th )
)  <->  -.  ( th  ->  -.  ph ) )
3324, 32anbi12i 678 . . . . . . . . . 10  |-  ( ( ( th  -/\  ch )  /\  ( ( ph  -/\  th )  -/\  ( ph  -/\  th )
) )  <->  ( ( th  ->  -.  ch )  /\  -.  ( th  ->  -. 
ph ) ) )
3433notbii 287 . . . . . . . . 9  |-  ( -.  ( ( th  -/\  ch )  /\  ( ( ph  -/\  th )  -/\  ( ph  -/\  th )
) )  <->  -.  (
( th  ->  -.  ch )  /\  -.  ( th  ->  -.  ph ) ) )
35 df-nan 1288 . . . . . . . . 9  |-  ( ( ( th  -/\  ch )  -/\  ( ( ph  -/\  th )  -/\  ( ph  -/\  th )
) )  <->  -.  (
( th  -/\  ch )  /\  ( ( ph  -/\  th )  -/\  ( ph  -/\  th )
) ) )
36 iman 413 . . . . . . . . 9  |-  ( ( ( th  ->  -.  ch )  ->  ( th 
->  -.  ph ) )  <->  -.  ( ( th  ->  -. 
ch )  /\  -.  ( th  ->  -.  ph )
) )
3734, 35, 363bitr4i 268 . . . . . . . 8  |-  ( ( ( th  -/\  ch )  -/\  ( ( ph  -/\  th )  -/\  ( ph  -/\  th )
) )  <->  ( ( th  ->  -.  ch )  ->  ( th  ->  -.  ph ) ) )
3821, 37anbi12i 678 . . . . . . 7  |-  ( ( ( ta  -/\  ( ta  -/\  ta ) )  /\  ( ( th 
-/\  ch )  -/\  (
( ph  -/\  th )  -/\  ( ph  -/\  th )
) ) )  <->  ( ( ta  ->  ( ta  /\  ta ) )  /\  (
( th  ->  -.  ch )  ->  ( th 
->  -.  ph ) ) ) )
3915, 38xchbinx 301 . . . . . 6  |-  ( ( ( ta  -/\  ( ta  -/\  ta ) ) 
-/\  ( ( th 
-/\  ch )  -/\  (
( ph  -/\  th )  -/\  ( ph  -/\  th )
) ) )  <->  -.  (
( ta  ->  ( ta  /\  ta ) )  /\  ( ( th 
->  -.  ch )  -> 
( th  ->  -.  ph ) ) ) )
4014, 39anbi12i 678 . . . . 5  |-  ( ( ( ph  -/\  ( ch  -/\  ps ) )  /\  ( ( ta 
-/\  ( ta  -/\  ta ) )  -/\  (
( th  -/\  ch )  -/\  ( ( ph  -/\  th )  -/\  ( ph  -/\  th )
) ) ) )  <-> 
( ( ph  ->  ( ch  /\  ps )
)  /\  -.  (
( ta  ->  ( ta  /\  ta ) )  /\  ( ( th 
->  -.  ch )  -> 
( th  ->  -.  ph ) ) ) ) )
4140notbii 287 . . . 4  |-  ( -.  ( ( ph  -/\  ( ch  -/\  ps ) )  /\  ( ( ta 
-/\  ( ta  -/\  ta ) )  -/\  (
( th  -/\  ch )  -/\  ( ( ph  -/\  th )  -/\  ( ph  -/\  th )
) ) ) )  <->  -.  ( ( ph  ->  ( ch  /\  ps )
)  /\  -.  (
( ta  ->  ( ta  /\  ta ) )  /\  ( ( th 
->  -.  ch )  -> 
( th  ->  -.  ph ) ) ) ) )
42 iman 413 . . . 4  |-  ( ( ( ph  ->  ( ch  /\  ps ) )  ->  ( ( ta 
->  ( ta  /\  ta ) )  /\  (
( th  ->  -.  ch )  ->  ( th 
->  -.  ph ) ) ) )  <->  -.  (
( ph  ->  ( ch 
/\  ps ) )  /\  -.  ( ( ta  ->  ( ta  /\  ta )
)  /\  ( ( th  ->  -.  ch )  ->  ( th  ->  -.  ph ) ) ) ) )
4341, 42bitr4i 243 . . 3  |-  ( -.  ( ( ph  -/\  ( ch  -/\  ps ) )  /\  ( ( ta 
-/\  ( ta  -/\  ta ) )  -/\  (
( th  -/\  ch )  -/\  ( ( ph  -/\  th )  -/\  ( ph  -/\  th )
) ) ) )  <-> 
( ( ph  ->  ( ch  /\  ps )
)  ->  ( ( ta  ->  ( ta  /\  ta ) )  /\  (
( th  ->  -.  ch )  ->  ( th 
->  -.  ph ) ) ) ) )
448, 43mpbir 200 . 2  |-  -.  (
( ph  -/\  ( ch 
-/\  ps ) )  /\  ( ( ta  -/\  ( ta  -/\  ta )
)  -/\  ( ( th  -/\  ch )  -/\  ( ( ph  -/\  th )  -/\  ( ph  -/\  th )
) ) ) )
45 df-nan 1288 . 2  |-  ( ( ( ph  -/\  ( ch  -/\  ps ) ) 
-/\  ( ( ta 
-/\  ( ta  -/\  ta ) )  -/\  (
( th  -/\  ch )  -/\  ( ( ph  -/\  th )  -/\  ( ph  -/\  th )
) ) ) )  <->  -.  ( ( ph  -/\  ( ch  -/\  ps ) )  /\  ( ( ta 
-/\  ( ta  -/\  ta ) )  -/\  (
( th  -/\  ch )  -/\  ( ( ph  -/\  th )  -/\  ( ph  -/\  th )
) ) ) ) )
4644, 45mpbir 200 1  |-  ( (
ph  -/\  ( ch  -/\  ps ) )  -/\  (
( ta  -/\  ( ta  -/\  ta ) ) 
-/\  ( ( th 
-/\  ch )  -/\  (
( 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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