This may seem like a dumb question, and it certainly seems dumb to me asking it, but I'm running into a contradiction. I'm looking at the problem of finding a statement $\phi$ such that $\psi$ and $\chi$ together are equivalent to $\phi$ but neither one on their own implies $\phi$:
$\phi$ $\longleftrightarrow$ ($\psi$ $\land$ $\chi$)
$\lnot$ ($\psi$ $\rightarrow$ $\phi$)
$\lnot$ ($\chi$ $\rightarrow$ $\phi$)
The problem is, when I evaluate the conjunction of all these statements in propositional calculus, it adds up to a contradiction, which seems a bit strange to me. I didn't expect that you couldn't compose a statement of two smaller statements that on their own don't imply $\phi$ but together are equal to it. I expect I'm doing something wrong in my formulation of my problem, or somehow I evaluated the statements wrong.
Say for example $\phi$ is the Axiom of Choice and $\psi$ is the Axiom of Dependent Choice, this implies there's no statement $\chi$ that doesn't fully imply the Axiom of Choice on its own to satisfy the first statement. I feel I must be doing something wrong, because it seems weird that I can't take two weak statements to make a strong statement. It seems so wrong I'm more than a little embarrassed asking this.
I don't think there's any magical translation that lets you turn invalid propositional statements into valid first order statements. Is there something I'm missing here? The conclusion seems surprising, but I feel like I'm forgetting something obvious.