Predicate logic proof (some a are b, some a are c, therefore there exists some c) For the syllogism:
Some A are B
Some A are C
------------
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?

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.

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