Theorem luklem2 1424

Description: Used to rederive standard propositional axioms from Lukasiewicz'. (Contributed by NM, 22-Dec-2002.) (Proof modification is discouraged.) (New usage is discouraged.)

Assertion

Ref	Expression
luklem2	⊢ ((φ → ¬ ψ) → (((φ → χ) → θ) → (ψ → θ)))

Proof of Theorem luklem2

Step	Hyp	Ref	Expression
1		luk-1 1420	. . 3 ⊢ ((φ → ¬ ψ) → ((¬ ψ → χ) → (φ → χ)))
2		luk-3 1422	. . . 4 ⊢ (ψ → (¬ ψ → χ))
3		luk-1 1420	. . . 4 ⊢ ((ψ → (¬ ψ → χ)) → (((¬ ψ → χ) → (φ → χ)) → (ψ → (φ → χ))))
4	2, 3	ax-mp 5	. . 3 ⊢ (((¬ ψ → χ) → (φ → χ)) → (ψ → (φ → χ)))
5	1, 4	luklem1 1423	. 2 ⊢ ((φ → ¬ ψ) → (ψ → (φ → χ)))
6		luk-1 1420	. 2 ⊢ ((ψ → (φ → χ)) → (((φ → χ) → θ) → (ψ → θ)))
7	5, 6	luklem1 1423	1 ⊢ ((φ → ¬ ψ) → (((φ → χ) → θ) → (ψ → θ)))

This proof depends on syntax axioms:  ¬ wn 3   → wi 4

This proof depends on axioms:  ax-mp 5  ax-meredith 1406

This theorem is used by:  luklem3  1425  luklem6  1428  ax3  1433