Theorem tbwlem1
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.)
Conclusion
\( \vdash (\varphi \rightarrow \psi \rightarrow \chi) \rightarrow \psi \rightarrow \varphi \rightarrow \chi \)
Proof
l0
tbw-ax1
\( \vdash (\varphi \rightarrow \psi \rightarrow \chi) \rightarrow ((\psi \rightarrow \chi) \rightarrow \chi) \rightarrow \varphi \rightarrow \chi \)
l1
tbw-ax2
\( \vdash \psi \rightarrow (\psi \rightarrow \chi) \rightarrow \psi \)
tbw-ax1
\( \vdash ((\psi \rightarrow \chi) \rightarrow \psi) \rightarrow (\psi \rightarrow \chi) \rightarrow (\psi \rightarrow \chi) \rightarrow \chi \)
l3
tbwsyl l1 _
\( \vdash \psi \rightarrow (\psi \rightarrow \chi) \rightarrow (\psi \rightarrow \chi) \rightarrow \chi \)
l4
tbw-ax1
\( \vdash ((\psi \rightarrow \chi) \rightarrow (\psi \rightarrow \chi) \rightarrow \chi) \rightarrow (((\psi \rightarrow \chi) \rightarrow \chi) \rightarrow \chi) \rightarrow (\psi \rightarrow \chi) \rightarrow \chi \)
tbw-ax3
\( \vdash ((((\psi \rightarrow \chi) \rightarrow \chi) \rightarrow \chi) \rightarrow (\psi \rightarrow \chi) \rightarrow \chi) \rightarrow (\psi \rightarrow \chi) \rightarrow \chi \)
tbwsyl l4 _
\( \vdash ((\psi \rightarrow \chi) \rightarrow (\psi \rightarrow \chi) \rightarrow \chi) \rightarrow (\psi \rightarrow \chi) \rightarrow \chi \)
l7
tbwsyl l3 _
\( \vdash \psi \rightarrow (\psi \rightarrow \chi) \rightarrow \chi \)
tbw-ax1
\( \vdash (\psi \rightarrow (\psi \rightarrow \chi) \rightarrow \chi) \rightarrow (((\psi \rightarrow \chi) \rightarrow \chi) \rightarrow \varphi \rightarrow \chi) \rightarrow \psi \rightarrow \varphi \rightarrow \chi \)
ax-mp l7 _
\( \vdash (((\psi \rightarrow \chi) \rightarrow \chi) \rightarrow \varphi \rightarrow \chi) \rightarrow \psi \rightarrow \varphi \rightarrow \chi \)
tbwsyl l0 _