Is this a weaker version of Simonyi's Conjecture

Question

So, I'm aware of Simonyi's conjecture which says that if $\mathcal{A}, \mathcal{B} \subset \mathcal{P}(n)$ satisfy the conditions: $$\forall A,A'\in\mathcal{A} \mbox{ and } \forall B, B' \in\mathcal{B}, (A \backslash B)=(A'\backslash B') \implies A=A' $$ $$\forall A,A'\in\mathcal{A} \mbox{ and } \forall B, B' \in\mathcal{B}, (B \backslash A)=(B'\backslash A') \implies B=B' $$ Then $|\mathcal{A}||\mathcal{B}|\leq2^{n}$. But, my teacher mentioned the other day something like if $\mathcal{A}, \mathcal{B} \subset \mathcal{P}(n)$ are such that $|A \cap B|$ is always even, then we also had that $|\mathcal{A}||\mathcal{B}|\leq2^{n}$. Firstly, is this true? And, secondly, is this a weaker version of the conditions in Simonyi's conjecture?