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Suppose the following have been established or assumed:

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This is as far as I can get:

Assume p.
It follow from (Axiom 1) that ~y.
It follows from (Thrm. 1) that ~z.

Are you sure you copied down the Axioms and Theorems correctly?

If we also assumed ~x, then
it follows from (Thrm. 3) that ~r.
Finally it follows from (Axiom 2) that q.

Therefor, (p and ~x) -> q