Theorem merlem4

Step 8 of Meredith's proof of Lukasiewicz axioms from his sole axiom. (Contributed by NM, 14-Dec-2002.) (Proof modification is discouraged.) (New usage is discouraged.)

Conclusion
\( \vdash \tau \rightarrow (\tau \rightarrow \varphi) \rightarrow \theta \rightarrow \varphi \)
Proof
l0
meredith
\( \vdash ((((\varphi \rightarrow \varphi) \rightarrow \neg \theta \rightarrow \neg \theta) \rightarrow \theta) \rightarrow \tau) \rightarrow (\tau \rightarrow \varphi) \rightarrow \theta \rightarrow \varphi \)
merlem3
\( \vdash (((((\varphi \rightarrow \varphi) \rightarrow \neg \theta \rightarrow \neg \theta) \rightarrow \theta) \rightarrow \tau) \rightarrow (\tau \rightarrow \varphi) \rightarrow \theta \rightarrow \varphi) \rightarrow \tau \rightarrow (\tau \rightarrow \varphi) \rightarrow \theta \rightarrow \varphi \)
ax-mp l0 _