Theorem tbwsyl 1469

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.)

Hypotheses

Ref	Expression
tbwsyl.1	⊢ (φ → ψ)
tbwsyl.2	⊢ (ψ → χ)

Assertion

Ref	Expression
tbwsyl	⊢ (φ → χ)

Proof of Theorem tbwsyl

Step	Hyp	Ref	Expression
1		tbwsyl.2	. 2 ⊢ (ψ → χ)
2		tbwsyl.1	. . 3 ⊢ (φ → ψ)
3		tbw-ax1 1465	. . 3 ⊢ ((φ → ψ) → ((ψ → χ) → (φ → χ)))
4	2, 3	ax-mp 5	. 2 ⊢ ((ψ → χ) → (φ → χ))
5	1, 4	ax-mp 5	1 ⊢ (φ → χ)

Syntax hints:   → wi 4

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

This theorem is referenced by:  tbwlem1  1470  tbwlem2  1471  tbwlem3  1472  tbwlem4  1473  tbwlem5  1474  re1luk2  1476