Theorem tbwsyl 1704

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 1700	. . 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  1705  tbwlem2  1706  tbwlem3  1707  tbwlem4  1708  tbwlem5  1709  re1luk2  1711  re1luk3  1712