Predicate logic proof (some a are b, some a are c, therefore there exists some c)

Question

For the syllogism:

Some A are B
Some A are C
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There exists C

Something like: My cake is pink, My cake is round, there exist things that are round

We get ?:

$\exists x (A(x) \land B(x)) \land \exists x(A(x) \land C(x)) \implies \exists x(C(x))$

How would I go about proving that predicate?

Answer 1

Here is a natural deduction proof using a Fitch-style proof checker:

Since $A$ is used to code $\forall$ in this tool, I replaced $A$ with $P$.

The proof uses in this order conjunction elimination, existential introduction and existential elimination.

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Kevin Klement's JavaScript/PHP Fitch-style natural deduction proof editor and checker http://proofs.openlogicproject.org/