Theorem euim

Description: Add unique existential quantifiers to an implication. Note the reversed implication in the antecedent. (Contributed by NM, 19-Oct-2005) (Proof shortened by Andrew Salmon, 14-Jun-2011) (Proof shortened by Wolf Lammen, 1-Oct-2023)

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

Proof

Step	Hyp	Ref	Expression
1		euimmo	$$⊢\forall x\left(\phi \to \psi \right)\to \left(\exists !x\psi \to {\exists }^{*}x\phi \right)$$
2		exmoeub	$$⊢\exists x\phi \to \left({\exists }^{*}x\phi \leftrightarrow \exists !x\phi \right)$$
3	2	biimpd	$$⊢\exists x\phi \to \left({\exists }^{*}x\phi \to \exists !x\phi \right)$$
4	1 3	sylan9r	$$⊢\left(\exists x\phi \wedge \forall x\left(\phi \to \psi \right)\right)\to \left(\exists !x\psi \to \exists !x\phi \right)$$