Write theorem conditions concisely

Question

Let $Z$ be a set, $x$ be some object.

Let the following statements hold (for some logical formulas $P,P_1,\dots,P_n$ and some logical formula $Q$):

1. $\forall z\in Z:(P(x) \Leftrightarrow Q(x,z))$

2. $(P_1(x)\Leftrightarrow P(x))\wedge\ldots\wedge(P_n(x)\Leftrightarrow P(x))$ for some natural $n$.

Question: How to formulate a theorem expressing truth of statements 1. and 2. concisely? (Note that $P,P_1,\dots,P_n,Q$ are relatively complex logical formulas and I prefer to write every of these formulas once, not repeat the same formula twice of more.)

Answer 1

If you are looking for a symbolic expression, there's not much hope; to express multiple equivalences, we don't have an option other than to use $\iff$, and to express that all $P_i$ are equivalent to $P$ we need to make a tree graph on $\{P, P_1, \ldots, P_n\}$, each connection being an equivalence. This is already accomplished by what you have, so we can't do any better.

If you're allowing a textual expression for your theorem, then we could try:

The following are equivalent:

• $P(x)$
• $P_i(x)$, for some $1 \le i \le n$
• $\exists z \in Z: Q(x,z)$