Here x cannot be a free variable of φ, while θ can be a predicate depending on it.
The following is easily established: In the first scenario, some x bound in the premise is reused in the conclusion, and it is generally not the a priori a that validates it.
Thirdly, now in contrast to the two points above, consider the case in which it is not known how to prove or reject φ.
More specifically, θ shall express that x is the index of a formal proof of some mathematical conjecture whose provability is not known.
In the proof calculus - like in the weak counterexample - a suitable x can only be given using more input tied to amenable φ.
Indeed, using violating models, it has been established that the premise φ → ∃x θ does not suffice for a generic proof of existence as granted by the principle.
[1] For existential-quantifier-free φ, theories over intuitionistic logic tend to be well behaved in regard to rules of this nature.
Hence, assuming the law of the excluded middle disjunction axiomatically, the principle is valid.
Constructively, one needs to provide an x such that one can demonstrate (then aided by φ assumed valid and so also ∃y θ for some y) that θ holds for that x. Classically, it suffices to draw the same conclusion of interest when starting from two hypotheticals about φ.
In the intuitionistic calculus, the finite form may be understood as expressing that information in the premise
, this reduces to the shorter but indeed equivalent so called Dirk Gently’s principle
The schema implies the strictly weaker excluded middle for negated propositions (WLEM) through the intuitionistic form of consequentia mirabilis.