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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

Proof of Theorem nic-ax
StepHypRef Expression
1 nannan 1300 . . . . 5
21biimpi 187 . . . 4
3 simpl 444 . . . . 5
43imim2i 14 . . . 4
5 imnan 412 . . . . . . 7
6 df-nan 1297 . . . . . . 7
75, 6bitr4i 244 . . . . . 6
8 con3 128 . . . . . . . 8
98imim2d 50 . . . . . . 7
10 imnan 412 . . . . . . . 8
11 con2b 325 . . . . . . . 8
12 df-nan 1297 . . . . . . . 8
1310, 11, 123bitr4ri 270 . . . . . . 7
149, 13syl6ibr 219 . . . . . 6
157, 14syl5bir 210 . . . . 5
16 nanim 1301 . . . . 5
1715, 16sylib 189 . . . 4
182, 4, 173syl 19 . . 3
19 pm4.24 625 . . . . 5
2019biimpi 187 . . . 4
21 nannan 1300 . . . 4
2220, 21mpbir 201 . . 3
2318, 22jctil 524 . 2
24 nannan 1300 . 2
2523, 24mpbir 201 1
 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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