Proving (∀x.ϕ) →(∃x.ϕ) using Natural Deduction?

[badwolf23]

Hello, I have no idea how to solve this proof and would actually appreciate your help. I cannot use soundness or completeness.

⊢ (∀x.ϕ) →(∃x.ϕ)


Thanks

 

[MLP]

Hello, I have no idea how to solve this proof and would actually appreciate your help. I cannot use soundness or completeness.

⊢ (∀x.ϕ) →(∃x.ϕ)Thanks

It will depend on the particular deductive apparatus. You might be using a set of axioms together with a set of rules of inference, or you might be using a system of natural deduction, etc. Let's suppose it's a natural deduction system.

Assume your antecedent. Often you will have a rule of universal instantiation that allows you to go from the assumed antecedent to an instance, then a rule that let's you go from the instance to the existential generalization of the instance (which is the consequent). Then an application of conditionalization gets you home. Does that help?