Exercise 5, § 5

Tag 03GQ, permanent. Cite it as [Bourbaki, Tag 03GQ].

Theory of Sets, I, 59

Let $R$ , $S$ , $T$ be relations in $\mathscr{T}$ and let $x$ be a letter. If $R$ is functional in $x$ in $\mathscr{T}$ , show that the following relations are theorems in $\mathscr{T}$ :

(\text{not } (\exists x)(R \text{ and } S)) \Leftrightarrow (\exists x)(R \text{ and } (\text{not } S)),

(\exists x)(R \text{ and } (S \text{ and } T)) \Leftrightarrow (\exists x)(R \text{ and } S) \text{ and } (\exists x)(R \text{ and } T)),

(\exists x)(R \text{ and } (S \text{ or } T)) \Leftrightarrow ((\exists x)(R \text{ and } S) \text{ or } (\exists x)(R \text{ and } T)).

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