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	|- ((ph -/\ (ch -/\ ps)) -/\ ((ta -/\ (ta -/\ ta)) -/\ ((th -/\ ch) -/\ ((ph -/\ th) -/\ (ph -/\ th)))))

Proof of Theorem nic-ax

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

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