I try to undestand the generalized tautology theorem (not sure wether it's the correct english expression) and its proof.

$\displaystyle \phi_1,...,\phi_n \models \phi \Rightarrow \phi_1,...,\phi_n \vdash \phi$

Doesn't it simply state the completeness of the axiom system (if it holds forall phi1..n)?

In the proof there is a curious step. It states that viamodus ponensyou can transform the first in the second statement:

$\displaystyle \phi_1\vdash\phi_1 \rightarrow ...\rightarrow \phi_n \rightarrow \phi$

$\displaystyle \phi_1\vdash\phi_2 \rightarrow ...\rightarrow \phi_n \rightarrow \phi$

I undestand the transformation, but I don't see modus ponens applied here. Anyone knows?

ps: Btw, where can I look up translations of mathematical expressions (e.g. completeness/Vollständigkeit,...) for English/German? I'm not used doing mathematics in English