Consider the following argument.

(i) ‘Fa’ and ‘a’ (or ‘-Fa’) are contradictories.

(ii) So one or other of ‘Fa’ and ‘a’ must be true.

(iii) Now ‘Fa’ entails ‘(x)Fx’,

(iv) And ‘a’ entails ‘(x)x’,

(v) So one or other of ‘(x)Fx’ and ‘(x)x’ must be true.

(vi) But both of ‘(x)Fx’ and ‘(x)x’ say there is something.

(vii) So, either way, there is something, i.e. something exists.