[FOM] Proof Theory Query

[joeshipman@aol.com]

Let T be a recursively axiomatizable first-order theory in a language 
with countably infinitely many constant symbols c0, c1, ....

Consider the following properties:

P1: Whenever ExPhi(x) is in T, there exists i such that Phi(c_i) is in 
T.

P2: Whenever AxPhi(x)-->Psi is in T, there exist finitely many 
constants c_i1, c_i2, ..., c_i_n such that 
((Phi(c_i1)&Phi(c_i2)&...&Phi(c_i_n))-->Psi) is in T.

What are these properties called, and can a theory have one but not the 
other?

-- JS