- The Father = God
- The Son = God
- The Holy Spirit = God
- The Father ≠ The Son ≠ The Holy Spirit
The Principle of the Identity of Indiscernibles states that for any two lists of predicates belonging to some beings hypothetically distinguished as A and B respectively, A ≡ B if and only if the lists of predicates are identical (containing all and only the same predicates). More than that, it implies that A is B and B is A. In other words, A and B designate the very same being, and cannot not designate the same being.
The doctrine of the order of procession in the Trinity allows us to make a distinction, say, between the Father and the Son. The Father is the fountainhead of the Trinity, as the Son comes from the Father, but not the Father from the Son (and this causation is efficient and non-successive). The Father and Son can be distinguished, in other words, precisely because there are prepositions which distinguish the one from the other (these prepositions or relations can be translated to predicates).
Now, according to those who deny the Filioque clause in the Nicene Creed as it is recited in the Latin west, such as some Eastern Orthodox Christians, the Holy Spirit and the Son relate to the Father in an identical way (since the Holy Spirit proceeds from the Father, and the Son proceeds from the Father, and the Holy Spirit proceeds from the Father without the Son, and the Son proceeds from the Father without the Holy Spirit). Therefore, there is no way to distinguish the Holy Spirit from the Son. They have all and only the same predicates, because even the relations which distinguish them from the Father do not help to distinguish them from each other. Therefore, if one denies the Filioque, it seems as though they cannot consistently be Trinitarians.
Perhaps they may respond by distinguishing the Son from the Holy Spirit by suggesting that the Son is incarnated whereas the Holy Spirit is not, or by appealing to differences between them insofar as they relate to the world (prepositions given creation). However, this would mean that God only became truly Trinitarian, or rather more formally that God becomes truly Trinitarian, at the moment of, or given, creation. That is not good enough for the Christian. God must be essentially Trinitarian, even in the absence of creation (since the act of creating was itself freely chosen by God such that he may not have created).
Therefore, without the Filioque (or some relevantly similar distinction between the Son and Holy Spirit) one cannot consistently distinguish three persons in God.