I know i can represent that as ~(Av1)~(v1=v1)
A = for all
E= there exist.
Is this on the right track?
: A(v1) initial sequent note the A is a formula
As emakrov said, this depends on what proof system you're using.
In one system, such a proof is simple:
1. Ax x=x ... axiom of identity theory
2. x=x ... universal instantiation
3. Ex x=x ... existential generalization
First, it's not propositional, but first-order calculus since your formulas include an existential quantifier and a binary equality predicate.
Post numbers are locates in the Bali Hai-colored bar above each post. Post #3 is located here.
Of various deductive system that I know, introduction and elimination occur in natural deduction. Usually ND is formulated so that derivations consist of formulas; however, there is also a sequent presentation where all open assumptions are collected in the left-hand side of sequents. In any case, MoeBlee's outline is pretty close.
To have a precise derivation, please provide the rules of universal elimination, existential introduction and describe how equality is incorporated into the system.