There are a couple of non-universal notations here. What is a "Regular Φ axiom system", namely what does Φ mean? And is the formula stating consistency of , something that is also denoted or ?
Presumably, says something "there is no -derivation of 0=1". If it is false (in the standard model ), then there exists, in fact, a -derivation of 0=1. Then is contradictory and therefore derives everything. However, it is proved that does not prove .