I'm currently trying to solve the following:
(∃x(¬A(x))) → [∀x (A(x)) → B(z) ]
using only the rules of predicate and propositional calculus. I've had a few stabs at the problem. My chief idea has to do with the A(x) statements. I understand that if there exists at least one false A(x), then A(x) is not all true. I understand that falsities can imply anything. How do I join these two ideas together with the proper formal logical notation?
For example, I first assume:
[∀x (A(x)) → B(z) ] (=> E) [∀x (A(x))] (assume) A(a) (∀E) F
So I have proved it's false. Now what?
I realise that there is a very good chance I am barking up the wrong tree but I cannot see any other way to prove this. B(z) is totally unrelated to A(z) and while existential elimination will let me arrive at a "q" statement, I do not see how I can get "q" from the other two terms.