Theorem tbwlem1 1470

Description: Used to rederive the Lukasiewicz axioms from Tarski-Bernays-Wajsberg'. (Contributed by Anthony Hart, 16-Aug-2011.) (Proof modification is discouraged.) (New usage is discouraged.)

Assertion

Ref	Expression
tbwlem1	⊢ ((φ → (ψ → χ)) → (ψ → (φ → χ)))

Proof of Theorem tbwlem1

Step	Hyp	Ref	Expression
1		tbw-ax2 1466	. . . 4 ⊢ (ψ → ((ψ → χ) → ψ))
2		tbw-ax1 1465	. . . 4 ⊢ (((ψ → χ) → ψ) → ((ψ → χ) → ((ψ → χ) → χ)))
3	1, 2	tbwsyl 1469	. . 3 ⊢ (ψ → ((ψ → χ) → ((ψ → χ) → χ)))
4		tbw-ax1 1465	. . . 4 ⊢ (((ψ → χ) → ((ψ → χ) → χ)) → ((((ψ → χ) → χ) → χ) → ((ψ → χ) → χ)))
5		tbw-ax3 1467	. . . 4 ⊢ (((((ψ → χ) → χ) → χ) → ((ψ → χ) → χ)) → ((ψ → χ) → χ))
6	4, 5	tbwsyl 1469	. . 3 ⊢ (((ψ → χ) → ((ψ → χ) → χ)) → ((ψ → χ) → χ))
7	3, 6	tbwsyl 1469	. 2 ⊢ (ψ → ((ψ → χ) → χ))
8		tbw-ax1 1465	. 2 ⊢ ((φ → (ψ → χ)) → (((ψ → χ) → χ) → (φ → χ)))
9		tbw-ax1 1465	. 2 ⊢ ((ψ → ((ψ → χ) → χ)) → ((((ψ → χ) → χ) → (φ → χ)) → (ψ → (φ → χ))))
10	7, 8, 9	mpsyl 59	1 ⊢ ((φ → (ψ → χ)) → (ψ → (φ → χ)))

Syntax hints:   → wi 4

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8

This theorem is referenced by:  tbwlem2  1471  tbwlem4  1473  tbwlem5  1474  re1luk3  1477