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Theorem nic-ax 1447
Description: Nicod's axiom derived from the standard ones. See _Intro. to Math. Phil._ by B. Russell, p. 152. Like meredith 1413, the usual axioms can be derived from this and vice versa. Unlike meredith 1413, Nicod uses a different connective ('nand'), so another form of modus ponens must be used in proofs, e.g.  { nic-ax 1447, nic-mp 1445  } is equivalent to  { luk-1 1429, luk-2 1430, luk-3 1431, ax-mp 8  }. In a pure (standalone) treatment of Nicod's axiom, this theorem would be changed to an axiom ($a statement). (Contributed by Jeff Hoffman, 19-Nov-2007.) (Proof modification is discouraged.) (New usage is discouraged.)
Assertion
Ref Expression
nic-ax  |-  ( (
ph  -/\  ( ch  -/\  ps ) )  -/\  (
( ta  -/\  ( ta  -/\  ta ) ) 
-/\  ( ( th 
-/\  ch )  -/\  (
( ph  -/\  th )  -/\  ( ph  -/\  th )
) ) ) )

Proof of Theorem nic-ax
StepHypRef Expression
1 nannan 1300 . . . . 5  |-  ( (
ph  -/\  ( ch  -/\  ps ) )  <->  ( ph  ->  ( ch  /\  ps ) ) )
21biimpi 187 . . . 4  |-  ( (
ph  -/\  ( ch  -/\  ps ) )  ->  ( ph  ->  ( ch  /\  ps ) ) )
3 simpl 444 . . . . 5  |-  ( ( ch  /\  ps )  ->  ch )
43imim2i 14 . . . 4  |-  ( (
ph  ->  ( ch  /\  ps ) )  ->  ( ph  ->  ch ) )
5 imnan 412 . . . . . . 7  |-  ( ( th  ->  -.  ch )  <->  -.  ( th  /\  ch ) )
6 df-nan 1297 . . . . . . 7  |-  ( ( th  -/\  ch )  <->  -.  ( th  /\  ch ) )
75, 6bitr4i 244 . . . . . 6  |-  ( ( th  ->  -.  ch )  <->  ( th  -/\  ch )
)
8 con3 128 . . . . . . . 8  |-  ( (
ph  ->  ch )  -> 
( -.  ch  ->  -. 
ph ) )
98imim2d 50 . . . . . . 7  |-  ( (
ph  ->  ch )  -> 
( ( th  ->  -. 
ch )  ->  ( th  ->  -.  ph ) ) )
10 imnan 412 . . . . . . . 8  |-  ( (
ph  ->  -.  th )  <->  -.  ( ph  /\  th ) )
11 con2b 325 . . . . . . . 8  |-  ( ( th  ->  -.  ph )  <->  (
ph  ->  -.  th )
)
12 df-nan 1297 . . . . . . . 8  |-  ( (
ph  -/\  th )  <->  -.  ( ph  /\  th ) )
1310, 11, 123bitr4ri 270 . . . . . . 7  |-  ( (
ph  -/\  th )  <->  ( th  ->  -.  ph ) )
149, 13syl6ibr 219 . . . . . 6  |-  ( (
ph  ->  ch )  -> 
( ( th  ->  -. 
ch )  ->  ( ph  -/\  th ) ) )
157, 14syl5bir 210 . . . . 5  |-  ( (
ph  ->  ch )  -> 
( ( th  -/\  ch )  ->  ( ph  -/\  th )
) )
16 nanim 1301 . . . . 5  |-  ( ( ( th  -/\  ch )  ->  ( ph  -/\  th )
)  <->  ( ( th 
-/\  ch )  -/\  (
( ph  -/\  th )  -/\  ( ph  -/\  th )
) ) )
1715, 16sylib 189 . . . 4  |-  ( (
ph  ->  ch )  -> 
( ( th  -/\  ch )  -/\  ( ( ph  -/\  th )  -/\  ( ph  -/\  th )
) ) )
182, 4, 173syl 19 . . 3  |-  ( (
ph  -/\  ( ch  -/\  ps ) )  ->  (
( th  -/\  ch )  -/\  ( ( ph  -/\  th )  -/\  ( ph  -/\  th )
) ) )
19 pm4.24 625 . . . . 5  |-  ( ta  <->  ( ta  /\  ta )
)
2019biimpi 187 . . . 4  |-  ( ta 
->  ( ta  /\  ta ) )
21 nannan 1300 . . . 4  |-  ( ( ta  -/\  ( ta  -/\ 
ta ) )  <->  ( ta  ->  ( ta  /\  ta ) ) )
2220, 21mpbir 201 . . 3  |-  ( ta 
-/\  ( ta  -/\  ta ) )
2318, 22jctil 524 . 2  |-  ( (
ph  -/\  ( ch  -/\  ps ) )  ->  (
( ta  -/\  ( ta  -/\  ta ) )  /\  ( ( th 
-/\  ch )  -/\  (
( ph  -/\  th )  -/\  ( ph  -/\  th )
) ) ) )
24 nannan 1300 . 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 )
) ) ) ) )
2523, 24mpbir 201 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 359    -/\ wnan 1296
This theorem is referenced by:  nic-imp  1449  nic-idlem1  1450  nic-idlem2  1451  nic-id  1452  nic-swap  1453  nic-luk1  1465  lukshef-ax1  1468
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 178  df-an 361  df-nan 1297
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