Theorem luklem2 662

Description: Lemma for rederiving standard propositional axioms from Lukasiewicz'.

Assertion

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

Proof of Theorem luklem2

Step	Hyp	Ref	Expression
1		luk-1 658	. . 3 ⊢ ((φ → ¬ ψ) → ((¬ ψ → χ) → (φ → χ)))
2		luk-3 660	. . . 4 ⊢ (ψ → (¬ ψ → χ))
3		luk-1 658	. . . 4 ⊢ ((ψ → (¬ ψ → χ)) → (((¬ ψ → χ) → (φ → χ)) → (ψ → (φ → χ))))
4	2, 3	ax-mp 6	. . 3 ⊢ (((¬ ψ → χ) → (φ → χ)) → (ψ → (φ → χ)))
5	1, 4	luklem1 661	. 2 ⊢ ((φ → ¬ ψ) → (ψ → (φ → χ)))
6		luk-1 658	. 2 ⊢ ((ψ → (φ → χ)) → (((φ → χ) → θ) → (ψ → θ)))
7	5, 6	luklem1 661	1 ⊢ ((φ → ¬ ψ) → (((φ → χ) → θ) → (ψ → θ)))

Syntax hints:  ¬ wn 1   → wi 2

This theorem is referenced by:  luklem3 663  luklem6 666  ax3 671

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

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