ph |- ph is not an axiom by your definition, as it has the $e statement ph.
>


Here "|-" is used loosely to separate the stack and the main statement. It 
belongs to a $e statement.
It is meaningful in a formalism where the stack is appended to the left of 
the main statement.

In full it would be:

hp $a [ Ga , ph ] |- ph

To separate $e statements and $p statements, I think metamath people use 
"&" and "=>". Look at the
theorems page for instances of this convention.
 
-- 
FL

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