Three examples come to mind, all with $P=ZF$ or some other base theory.

1. Fermat / Heath-Brown / Zagier:

• $Q$ = "every prime of the form $4n+1$ is the sum of two squares";
• $X$ = "the map $$(x,y,z)\mapsto \begin{cases} (x+2z,~z,~y-x-z),\quad \textrm{if}\,\,\, x < y-z \\ (2y-x,~y,~x-y+z),\quad \textrm{if}\,\,\, y-z < x < 2y\\ (x-2y,~x-y+z,~y),\quad \textrm{if}\,\,\, x > 2y \end{cases} $$ is an involution with an odd number of fixed points on the set $\{(x,y,z)\in\mathbb{N}^3:x^2+4yz=p\}$.

2. Hadamard / de la Vallee Poussin / Newman / Zagier:

• $Q$ = the prime number theorem;
• $X_1,\ldots,X_7$ = Zagier's statements $I-VI$ and the Analytic Theorem.

3. Lebesgue / Caratheodory:

• $Q$ = the Lebesgue-measurable sets are closed under countable unions;
• $X$ = the Caratheodory-measurable sets are closed under countable unions.

I think giving the statements $X$ and asking for proofs could be a reasonable homework or group project for a graduate class in Number Theory, Complex Analysis, or Real Analysis, which is saying something given the significance of the $Q$.