Theorem alseueu

Description: "The ph is ps " implies that exactly one thing is both ph and ps . This is the half of dfalseu2 that drops the universal conjunct; it does not reverse, so E! x ( ph /\ ps ) cannot be used in place of an "all some one" statement. (Contributed by David A. Wheeler, 21-Jul-2026)

	Ref	Expression
Assertion	alseueu	$$⊢\forall \exists !x\left(\phi \to \psi \right)\to \exists !x\left(\phi \wedge \psi \right)$$

Proof

Step	Hyp	Ref	Expression
1		dfalseu2	$$⊢\forall \exists !x\left(\phi \to \psi \right)\leftrightarrow \left(\forall x\left(\phi \to \psi \right)\wedge \exists !x\left(\phi \wedge \psi \right)\right)$$
2	1	simprbi	$$⊢\forall \exists !x\left(\phi \to \psi \right)\to \exists !x\left(\phi \wedge \psi \right)$$