Theorem luklem7 667

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

Assertion

Ref	Expression
luklem7	⊢ ((φ → (ψ → χ)) → (ψ → (φ → χ)))

Proof of Theorem luklem7

Step	Hyp	Ref	Expression
1		luk-1 658	. 2 ⊢ ((φ → (ψ → χ)) → (((ψ → χ) → χ) → (φ → χ)))
2		luklem5 665	. . . . 5 ⊢ (ψ → ((ψ → χ) → ψ))
3		luk-1 658	. . . . 5 ⊢ (((ψ → χ) → ψ) → ((ψ → χ) → ((ψ → χ) → χ)))
4	2, 3	luklem1 661	. . . 4 ⊢ (ψ → ((ψ → χ) → ((ψ → χ) → χ)))
5		luklem6 666	. . . 4 ⊢ (((ψ → χ) → ((ψ → χ) → χ)) → ((ψ → χ) → χ))
6	4, 5	luklem1 661	. . 3 ⊢ (ψ → ((ψ → χ) → χ))
7		luk-1 658	. . 3 ⊢ ((ψ → ((ψ → χ) → χ)) → ((((ψ → χ) → χ) → (φ → χ)) → (ψ → (φ → χ))))
8	6, 7	ax-mp 6	. 2 ⊢ ((((ψ → χ) → χ) → (φ → χ)) → (ψ → (φ → χ)))
9	1, 8	luklem1 661	1 ⊢ ((φ → (ψ → χ)) → (ψ → (φ → χ)))

Syntax hints:   → wi 2

This theorem is referenced by:  luklem8 668  ax2 670

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

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