The problems:

a)

for any

I'm confused about this one because I tried proof by contradiction, so:

(1)

by assumption

(2)

by assuming the negation of

where

is also written as

... and then isn't that already a contradiction?

b)

where x is not free in

This one I think it must involve axioms 4 and 5 somehow, but I guess that's obvious

c)

for any term

not involving x

And this one must involve axioms 4 and 6

Derive them using these (for your convenience and my LaTeX practice):

Axioms:

1:

2:

3:

4:

(

is a term freely substitutable for x in

)

5:

(where x is not free in

)

6:

7:

As well as Modus Ponens, Generalization, and the Deduction theorem

Any help would be really appreciated!