Theorem merco1lem12
Used to rederive the Tarski-Bernays-Wajsberg axioms from merco1.
(Contributed by Anthony Hart, 18-Sep-2011.)
(Proof modification is discouraged.) (New usage is discouraged.)
Conclusion
\( \vdash (\varphi \rightarrow \psi) \rightarrow ((\chi \rightarrow \varphi \rightarrow \tau) \rightarrow \varphi) \rightarrow \psi \)
Proof
l0
merco1lem3
\( \vdash (((\varphi \rightarrow \tau) \rightarrow ((\chi \rightarrow \varphi \rightarrow \tau) \rightarrow \varphi) \rightarrow \bot) \rightarrow \chi \rightarrow \bot) \rightarrow \chi \rightarrow \varphi \rightarrow \tau \)
merco1
\( \vdash ((((\varphi \rightarrow \tau) \rightarrow ((\chi \rightarrow \varphi \rightarrow \tau) \rightarrow \varphi) \rightarrow \bot) \rightarrow \chi \rightarrow \bot) \rightarrow \chi \rightarrow \varphi \rightarrow \tau) \rightarrow ((\chi \rightarrow \varphi \rightarrow \tau) \rightarrow \varphi) \rightarrow ((\chi \rightarrow \varphi \rightarrow \tau) \rightarrow \varphi) \rightarrow \varphi \)
l2
ax-mp l0 _
\( \vdash ((\chi \rightarrow \varphi \rightarrow \tau) \rightarrow \varphi) \rightarrow ((\chi \rightarrow \varphi \rightarrow \tau) \rightarrow \varphi) \rightarrow \varphi \)
merco1lem9
\( \vdash (((\chi \rightarrow \varphi \rightarrow \tau) \rightarrow \varphi) \rightarrow ((\chi \rightarrow \varphi \rightarrow \tau) \rightarrow \varphi) \rightarrow \varphi) \rightarrow ((\chi \rightarrow \varphi \rightarrow \tau) \rightarrow \varphi) \rightarrow \varphi \)
l4
ax-mp l2 _
\( \vdash ((\chi \rightarrow \varphi \rightarrow \tau) \rightarrow \varphi) \rightarrow \varphi \)
merco1lem11
\( \vdash (((\chi \rightarrow \varphi \rightarrow \tau) \rightarrow \varphi) \rightarrow \varphi) \rightarrow (((\psi \rightarrow \varphi) \rightarrow ((\chi \rightarrow \varphi \rightarrow \tau) \rightarrow \varphi) \rightarrow \bot) \rightarrow \bot) \rightarrow \varphi \)
l6
ax-mp l4 _
\( \vdash (((\psi \rightarrow \varphi) \rightarrow ((\chi \rightarrow \varphi \rightarrow \tau) \rightarrow \varphi) \rightarrow \bot) \rightarrow \bot) \rightarrow \varphi \)
merco1
\( \vdash ((((\psi \rightarrow \varphi) \rightarrow ((\chi \rightarrow \varphi \rightarrow \tau) \rightarrow \varphi) \rightarrow \bot) \rightarrow \bot) \rightarrow \varphi) \rightarrow (\varphi \rightarrow \psi) \rightarrow ((\chi \rightarrow \varphi \rightarrow \tau) \rightarrow \varphi) \rightarrow \psi \)
ax-mp l6 _