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Theorem nic-ax 1438

Description: Nicod's axiom derived from the standard ones. See _Intro. to Math. Phil._ by B. Russell, p. 152. Like meredith 1404, the usual axioms can be derived from this and vice versa. Unlike meredith 1404, Nicod uses a different connective ('nand'), so another form of modus ponens must be used in proofs, e.g. nic-ax 1438, nic-mp 1436 is equivalent to luk-1 1420, luk-2 1421, luk-3 1422, 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

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Proof of Theorem nic-ax

Step	Hyp	Ref	Expression
1		nannan 1291	. . . . 5
2	1	biimpi 186	. . . 4
3		simpl 443	. . . . 5
4	3	imim2i 13	. . . 4
5		imnan 411	. . . . . . 7
6		df-nan 1288	. . . . . . 7
7	5, 6	bitr4i 243	. . . . . 6
8		con3 126	. . . . . . . 8
9	8	imim2d 48	. . . . . . 7
10		imnan 411	. . . . . . . 8
11		con2b 324	. . . . . . . 8
12		df-nan 1288	. . . . . . . 8
13	10, 11, 12	3bitr4ri 269	. . . . . . 7
14	9, 13	syl6ibr 218	. . . . . 6
15	7, 14	syl5bir 209	. . . . 5
16		nanim 1292	. . . . 5
17	15, 16	sylib 188	. . . 4
18	2, 4, 17	3syl 18	. . 3
19		pm4.24 624	. . . . 5
20	19	biimpi 186	. . . 4
21		nannan 1291	. . . 4
22	20, 21	mpbir 200	. . 3
23	18, 22	jctil 523	. 2
24		nannan 1291	. 2
25	23, 24	mpbir 200	1

Colors of variables: wff setvar 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

This theorem is referenced by:  nic-imp  1440  nic-idlem1  1441  nic-idlem2  1442  nic-id  1443  nic-swap  1444  nic-luk1  1456  lukshef-ax1  1459

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