Theorem mercolem5
Used to rederive the Tarski-Bernays-Wajsberg axioms from merco2.
(Contributed by Anthony Hart, 16-Aug-2011.)
(Proof modification is discouraged.) (New usage is discouraged.)
Conclusion
\( \vdash \theta \rightarrow (\theta \rightarrow \varphi) \rightarrow \tau \rightarrow \chi \rightarrow \varphi \)
Proof
l0
merco2
\( \vdash ((\varphi \rightarrow \varphi) \rightarrow (\bot \rightarrow \varphi) \rightarrow \varphi) \rightarrow (\varphi \rightarrow \varphi) \rightarrow \varphi \rightarrow \varphi \rightarrow \varphi \)
l1
merco2
\( \vdash ((\varphi \rightarrow \varphi) \rightarrow (\bot \rightarrow \varphi) \rightarrow \theta) \rightarrow (\theta \rightarrow \varphi) \rightarrow \tau \rightarrow \chi \rightarrow \varphi \)
mercolem1
\( \vdash (((\varphi \rightarrow \varphi) \rightarrow (\bot \rightarrow \varphi) \rightarrow \theta) \rightarrow (\theta \rightarrow \varphi) \rightarrow \tau \rightarrow \chi \rightarrow \varphi) \rightarrow ((\bot \rightarrow \varphi) \rightarrow \theta) \rightarrow \theta \rightarrow (\theta \rightarrow \varphi) \rightarrow \tau \rightarrow \chi \rightarrow \varphi \)
l3
ax-mp l1 _
\( \vdash ((\bot \rightarrow \varphi) \rightarrow \theta) \rightarrow \theta \rightarrow (\theta \rightarrow \varphi) \rightarrow \tau \rightarrow \chi \rightarrow \varphi \)
l4
mercolem2
\( \vdash ((\theta \rightarrow (\theta \rightarrow \varphi) \rightarrow \tau \rightarrow \chi \rightarrow \varphi) \rightarrow \theta) \rightarrow (\bot \rightarrow \varphi) \rightarrow (\bot \rightarrow \varphi) \rightarrow \theta \)
merco2
\( \vdash (((\theta \rightarrow (\theta \rightarrow \varphi) \rightarrow \tau \rightarrow \chi \rightarrow \varphi) \rightarrow \theta) \rightarrow (\bot \rightarrow \varphi) \rightarrow (\bot \rightarrow \varphi) \rightarrow \theta) \rightarrow (((\bot \rightarrow \varphi) \rightarrow \theta) \rightarrow \theta \rightarrow (\theta \rightarrow \varphi) \rightarrow \tau \rightarrow \chi \rightarrow \varphi) \rightarrow (((\varphi \rightarrow \varphi) \rightarrow (\bot \rightarrow \varphi) \rightarrow \varphi) \rightarrow (\varphi \rightarrow \varphi) \rightarrow \varphi \rightarrow \varphi \rightarrow \varphi) \rightarrow (((\varphi \rightarrow \varphi) \rightarrow (\bot \rightarrow \varphi) \rightarrow \varphi) \rightarrow (\varphi \rightarrow \varphi) \rightarrow \varphi \rightarrow \varphi \rightarrow \varphi) \rightarrow \theta \rightarrow (\theta \rightarrow \varphi) \rightarrow \tau \rightarrow \chi \rightarrow \varphi \)
ax-mp l4 _
\( \vdash (((\bot \rightarrow \varphi) \rightarrow \theta) \rightarrow \theta \rightarrow (\theta \rightarrow \varphi) \rightarrow \tau \rightarrow \chi \rightarrow \varphi) \rightarrow (((\varphi \rightarrow \varphi) \rightarrow (\bot \rightarrow \varphi) \rightarrow \varphi) \rightarrow (\varphi \rightarrow \varphi) \rightarrow \varphi \rightarrow \varphi \rightarrow \varphi) \rightarrow (((\varphi \rightarrow \varphi) \rightarrow (\bot \rightarrow \varphi) \rightarrow \varphi) \rightarrow (\varphi \rightarrow \varphi) \rightarrow \varphi \rightarrow \varphi \rightarrow \varphi) \rightarrow \theta \rightarrow (\theta \rightarrow \varphi) \rightarrow \tau \rightarrow \chi \rightarrow \varphi \)
ax-mp l3 _
\( \vdash (((\varphi \rightarrow \varphi) \rightarrow (\bot \rightarrow \varphi) \rightarrow \varphi) \rightarrow (\varphi \rightarrow \varphi) \rightarrow \varphi \rightarrow \varphi \rightarrow \varphi) \rightarrow (((\varphi \rightarrow \varphi) \rightarrow (\bot \rightarrow \varphi) \rightarrow \varphi) \rightarrow (\varphi \rightarrow \varphi) \rightarrow \varphi \rightarrow \varphi \rightarrow \varphi) \rightarrow \theta \rightarrow (\theta \rightarrow \varphi) \rightarrow \tau \rightarrow \chi \rightarrow \varphi \)
ax-mp l0 _
\( \vdash (((\varphi \rightarrow \varphi) \rightarrow (\bot \rightarrow \varphi) \rightarrow \varphi) \rightarrow (\varphi \rightarrow \varphi) \rightarrow \varphi \rightarrow \varphi \rightarrow \varphi) \rightarrow \theta \rightarrow (\theta \rightarrow \varphi) \rightarrow \tau \rightarrow \chi \rightarrow \varphi \)
ax-mp l0 _