[FOM] Help with this statement!

[Zuhair Abdul Ghafoor Al-Johar]

Dear FoMers,

I made the following statement as means for proving Con(NF):-

"If T is a first order theory solely axiomatized (extra-logically) by a naive
like comprehension schema (Exist x for all y (y ϵ x <-> φ)), abbreviated as
(Exist x. set_of_all_φers(x)), for some syntactical characterization upon φ;
and if for a certain formula π (in one free variable) we have
T|- for all x(set_of_all_φers(x) -> π(x) & for all yϵx(π(y)))
per each formula φ used in comprehension; THEN this entails:
Con(T) ->Con(T+Ax(π(x)))".

The above statement is too general and hard to believe being true, yet I'm
having difficulty in finding a counter-example to it.

If this statement proves to be true, then it would provide a direct answer to
Con(NF)? to the positive! however I do highly doubt it.

Presenting a counter-example would help clear the fog!

Best Regards,

Zuhair