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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 ((φ (χ ψ)) ((τ (τ τ)) ((θ χ) ((φ θ) (φ θ)))))

Proof of Theorem nic-ax
StepHypRef Expression
1 nannan 1291 . . . . 5 ((φ (χ ψ)) ↔ (φ → (χ ψ)))
21biimpi 186 . . . 4 ((φ (χ ψ)) → (φ → (χ ψ)))
3 simpl 443 . . . . 5 ((χ ψ) → χ)
43imim2i 13 . . . 4 ((φ → (χ ψ)) → (φχ))
5 imnan 411 . . . . . . 7 ((θ → ¬ χ) ↔ ¬ (θ χ))
6 df-nan 1288 . . . . . . 7 ((θ χ) ↔ ¬ (θ χ))
75, 6bitr4i 243 . . . . . 6 ((θ → ¬ χ) ↔ (θ χ))
8 con3 126 . . . . . . . 8 ((φχ) → (¬ χ → ¬ φ))
98imim2d 48 . . . . . . 7 ((φχ) → ((θ → ¬ χ) → (θ → ¬ φ)))
10 imnan 411 . . . . . . . 8 ((φ → ¬ θ) ↔ ¬ (φ θ))
11 con2b 324 . . . . . . . 8 ((θ → ¬ φ) ↔ (φ → ¬ θ))
12 df-nan 1288 . . . . . . . 8 ((φ θ) ↔ ¬ (φ θ))
1310, 11, 123bitr4ri 269 . . . . . . 7 ((φ θ) ↔ (θ → ¬ φ))
149, 13syl6ibr 218 . . . . . 6 ((φχ) → ((θ → ¬ χ) → (φ θ)))
157, 14syl5bir 209 . . . . 5 ((φχ) → ((θ χ) → (φ θ)))
16 nanim 1292 . . . . 5 (((θ χ) → (φ θ)) ↔ ((θ χ) ((φ θ) (φ θ))))
1715, 16sylib 188 . . . 4 ((φχ) → ((θ χ) ((φ θ) (φ θ))))
182, 4, 173syl 18 . . 3 ((φ (χ ψ)) → ((θ χ) ((φ θ) (φ θ))))
19 pm4.24 624 . . . . 5 (τ ↔ (τ τ))
2019biimpi 186 . . . 4 (τ → (τ τ))
21 nannan 1291 . . . 4 ((τ (τ τ)) ↔ (τ → (τ τ)))
2220, 21mpbir 200 . . 3 (τ (τ τ))
2318, 22jctil 523 . 2 ((φ (χ ψ)) → ((τ (τ τ)) ((θ χ) ((φ θ) (φ θ)))))
24 nannan 1291 . 2 (((φ (χ ψ)) ((τ (τ τ)) ((θ χ) ((φ θ) (φ θ))))) ↔ ((φ (χ ψ)) → ((τ (τ τ)) ((θ χ) ((φ θ) (φ θ))))))
2523, 24mpbir 200 1 ((φ (χ ψ)) ((τ (τ τ)) ((θ χ) ((φ θ) (φ θ)))))
 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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