Theorem nic-ax 1239

Description: Nicod's axiom derived from the standard ones. See _Intro. to Math. Phil._ by B. Russell, p. 152. Like meredith 1200, the usual axioms can be derived from this and vice versa. Unlike meredith 1200, Nicod uses a different connective ('nand'), so another form of modus ponens must be used in proofs, e.g. { nic-ax 1239, nic-mp 1237 } is equivalent to { luk-1 1215, luk-2 1216, luk-3 1217, ax-mp 7 }. 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.)

Assertion

Ref	Expression
nic-ax	|- ((ph -/\ (ch -/\ ps)) -/\ ((ta -/\ (ta -/\ ta)) -/\ ((th -/\ ch) -/\ ((ph -/\ th) -/\ (ph -/\ th)))))

Proof of Theorem nic-ax

Step	Hyp	Ref	Expression
1		nic-justlem 1231	. . . . 5 |- ((ph -/\ (ch -/\ ps)) <-> (ph -> (ch /\ ps)))
2	1	biimpi 168	. . . 4 |- ((ph -/\ (ch -/\ ps)) -> (ph -> (ch /\ ps)))
3		simpl 346	. . . . 5 |- ((ch /\ ps) -> ch)
4	3	imim2i 11	. . . 4 |- ((ph -> (ch /\ ps)) -> (ph -> ch))
5		con3 110	. . . . . . . 8 |- ((ph -> ch) -> (-. ch -> -. ph))
6	5	imim2d 28	. . . . . . 7 |- ((ph -> ch) -> ((th -> -. ch) -> (th -> -. ph)))
7		imnan 261	. . . . . . . 8 |- ((ph -> -. th) <-> -. (ph /\ th))
8		con2b 182	. . . . . . . 8 |- ((th -> -. ph) <-> (ph -> -. th))
9		df-nand 1230	. . . . . . . 8 |- ((ph -/\ th) <-> -. (ph /\ th))
10	7, 8, 9	3bitr4ri 201	. . . . . . 7 |- ((ph -/\ th) <-> (th -> -. ph))
11	6, 10	syl6ibr 230	. . . . . 6 |- ((ph -> ch) -> ((th -> -. ch) -> (ph -/\ th)))
12		imnan 261	. . . . . . 7 |- ((th -> -. ch) <-> -. (th /\ ch))
13		df-nand 1230	. . . . . . 7 |- ((th -/\ ch) <-> -. (th /\ ch))
14	12, 13	bitr4i 193	. . . . . 6 |- ((th -> -. ch) <-> (th -/\ ch))
15	11, 14	syl5ibr 224	. . . . 5 |- ((ph -> ch) -> ((th -/\ ch) -> (ph -/\ th)))
16		nic-justim 1232	. . . . 5 |- (((th -/\ ch) -> (ph -/\ th)) <-> ((th -/\ ch) -/\ ((ph -/\ th) -/\ (ph -/\ th))))
17	15, 16	sylib 215	. . . 4 |- ((ph -> ch) -> ((th -/\ ch) -/\ ((ph -/\ th) -/\ (ph -/\ th))))
18	2, 4, 17	3syl 24	. . 3 |- ((ph -/\ (ch -/\ ps)) -> ((th -/\ ch) -/\ ((ph -/\ th) -/\ (ph -/\ th))))
19		pm4.24 479	. . . . 5 |- (ta <-> (ta /\ ta))
20	19	biimpi 168	. . . 4 |- (ta -> (ta /\ ta))
21		nic-justlem 1231	. . . 4 |- ((ta -/\ (ta -/\ ta)) <-> (ta -> (ta /\ ta)))
22	20, 21	mpbir 207	. . 3 |- (ta -/\ (ta -/\ ta))
23	18, 22	jctil 316	. 2 |- ((ph -/\ (ch -/\ ps)) -> ((ta -/\ (ta -/\ ta)) /\ ((th -/\ ch) -/\ ((ph -/\ th) -/\ (ph -/\ th)))))
24		nic-justlem 1231	. 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))))))
25	23, 24	mpbir 207	1 |- ((ph -/\ (ch -/\ ps)) -/\ ((ta -/\ (ta -/\ ta)) -/\ ((th -/\ ch) -/\ ((ph -/\ th) -/\ (ph -/\ th)))))

Syntax hints:  -. wn 2   -> wi 3   /\ wa 240   -/\ wnand 1229

This theorem is referenced by:  nic-imp 1241  nic-idlem1 1242  nic-idlem2 1243  nic-id 1244  nic-swap 1245  nic-luk1 1257  lukshef-ax1 14161

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7

This theorem depends on definitions:  df-bi 164  df-an 242  df-nand 1230

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