Theorem luklem7 1429

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

Proof of Theorem luklem7

Step	Hyp	Ref	Expression
1		luk-1 1420	. 2 ⊢ ((φ → (ψ → χ)) → (((ψ → χ) → χ) → (φ → χ)))
2		luklem5 1427	. . . . 5 ⊢ (ψ → ((ψ → χ) → ψ))
3		luk-1 1420	. . . . 5 ⊢ (((ψ → χ) → ψ) → ((ψ → χ) → ((ψ → χ) → χ)))
4	2, 3	luklem1 1423	. . . 4 ⊢ (ψ → ((ψ → χ) → ((ψ → χ) → χ)))
5		luklem6 1428	. . . 4 ⊢ (((ψ → χ) → ((ψ → χ) → χ)) → ((ψ → χ) → χ))
6	4, 5	luklem1 1423	. . 3 ⊢ (ψ → ((ψ → χ) → χ))
7		luk-1 1420	. . 3 ⊢ ((ψ → ((ψ → χ) → χ)) → ((((ψ → χ) → χ) → (φ → χ)) → (ψ → (φ → χ))))
8	6, 7	ax-mp 5	. 2 ⊢ ((((ψ → χ) → χ) → (φ → χ)) → (ψ → (φ → χ)))
9	1, 8	luklem1 1423	1 ⊢ ((φ → (ψ → χ)) → (ψ → (φ → χ)))

This proof depends on syntax axioms:   → wi 4

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

This theorem is used by:  luklem8  1430  ax2  1432