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Theorem nic-luk1 1456
 Description: Proof of luk-1 1420 from nic-ax 1438 and nic-mp 1436 (and definitions nic-dfim 1434 and nic-dfneg 1435). Note that the standard axioms ax-1 6, ax-2 7, and ax-3 8 are proved from the Lukasiewicz axioms by theorems ax1 1431, ax2 1432, and ax3 1433. (Contributed by Jeff Hoffman, 18-Nov-2007.) (Proof modification is discouraged.) (New usage is discouraged.)
Assertion
Ref Expression
nic-luk1 ((φψ) → ((ψχ) → (φχ)))

Proof of Theorem nic-luk1
StepHypRef Expression
1 nic-dfim 1434 . . . 4 (((φ (ψ ψ)) (φψ)) (((φ (ψ ψ)) (φ (ψ ψ))) ((φψ) (φψ))))
21nic-bi2 1454 . . 3 ((φψ) ((φ (ψ ψ)) (φ (ψ ψ))))
3 nic-ax 1438 . . . . . . 7 ((φ (ψ ψ)) ((τ (τ τ)) (((χ χ) ψ) ((φ (χ χ)) (φ (χ χ))))))
43nic-isw2 1446 . . . . . 6 ((φ (ψ ψ)) ((((χ χ) ψ) ((φ (χ χ)) (φ (χ χ)))) (τ (τ τ))))
54nic-idel 1449 . . . . 5 ((φ (ψ ψ)) ((((χ χ) ψ) ((φ (χ χ)) (φ (χ χ)))) (((χ χ) ψ) ((φ (χ χ)) (φ (χ χ))))))
6 nic-dfim 1434 . . . . . . . . 9 (((φ (χ χ)) (φχ)) (((φ (χ χ)) (φ (χ χ))) ((φχ) (φχ))))
76nic-bi1 1453 . . . . . . . 8 ((φ (χ χ)) ((φχ) (φχ)))
87nic-idbl 1451 . . . . . . 7 (((φχ) (φχ)) (((φ (χ χ)) (φ (χ χ))) ((φ (χ χ)) (φ (χ χ)))))
98nic-imp 1440 . . . . . 6 ((((χ χ) ψ) ((φ (χ χ)) (φ (χ χ)))) ((((φχ) (φχ)) ((χ χ) ψ)) (((φχ) (φχ)) ((χ χ) ψ))))
10 nic-dfim 1434 . . . . . . . . 9 (((ψ (χ χ)) (ψχ)) (((ψ (χ χ)) (ψ (χ χ))) ((ψχ) (ψχ))))
1110nic-bi2 1454 . . . . . . . 8 ((ψχ) ((ψ (χ χ)) (ψ (χ χ))))
12 nic-swap 1444 . . . . . . . 8 ((ψ (χ χ)) (((χ χ) ψ) ((χ χ) ψ)))
1311, 12nic-ich 1450 . . . . . . 7 ((ψχ) (((χ χ) ψ) ((χ χ) ψ)))
1413nic-imp 1440 . . . . . 6 ((((φχ) (φχ)) ((χ χ) ψ)) (((ψχ) ((φχ) (φχ))) ((ψχ) ((φχ) (φχ)))))
159, 14nic-ich 1450 . . . . 5 ((((χ χ) ψ) ((φ (χ χ)) (φ (χ χ)))) (((ψχ) ((φχ) (φχ))) ((ψχ) ((φχ) (φχ)))))
165, 15nic-ich 1450 . . . 4 ((φ (ψ ψ)) (((ψχ) ((φχ) (φχ))) ((ψχ) ((φχ) (φχ)))))
17 nic-dfim 1434 . . . . 5 ((((ψχ) ((φχ) (φχ))) ((ψχ) → (φχ))) ((((ψχ) ((φχ) (φχ))) ((ψχ) ((φχ) (φχ)))) (((ψχ) → (φχ)) ((ψχ) → (φχ)))))
1817nic-bi1 1453 . . . 4 (((ψχ) ((φχ) (φχ))) (((ψχ) → (φχ)) ((ψχ) → (φχ))))
1916, 18nic-ich 1450 . . 3 ((φ (ψ ψ)) (((ψχ) → (φχ)) ((ψχ) → (φχ))))
202, 19nic-ich 1450 . 2 ((φψ) (((ψχ) → (φχ)) ((ψχ) → (φχ))))
21 nic-dfim 1434 . . 3 ((((φψ) (((ψχ) → (φχ)) ((ψχ) → (φχ)))) ((φψ) → ((ψχ) → (φχ)))) ((((φψ) (((ψχ) → (φχ)) ((ψχ) → (φχ)))) ((φψ) (((ψχ) → (φχ)) ((ψχ) → (φχ))))) (((φψ) → ((ψχ) → (φχ))) ((φψ) → ((ψχ) → (φχ))))))
2221nic-bi1 1453 . 2 (((φψ) (((ψχ) → (φχ)) ((ψχ) → (φχ)))) (((φψ) → ((ψχ) → (φχ))) ((φψ) → ((ψχ) → (φχ)))))
2320, 22nic-mp 1436 1 ((φψ) → ((ψχ) → (φχ)))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ⊼ wnan 1287 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8 This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-nan 1288 This theorem is referenced by: (None)
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