Theorem euimOLD

Description: Obsolete version of euim as of 1-Oct-2023. (Contributed by NM, 19-Oct-2005) (Proof shortened by Andrew Salmon, 14-Jun-2011) (New usage is discouraged.) (Proof modification is discouraged.)

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

Proof

Step	Hyp	Ref	Expression
1		ax-1	$$⊢\exists x\phi \to \left(\exists !x\psi \to \exists x\phi \right)$$
2		euimmo	$$⊢\forall x\left(\phi \to \psi \right)\to \left(\exists !x\psi \to {\exists }^{*}x\phi \right)$$
3	1 2	anim12ii	$$⊢\left(\exists x\phi \wedge \forall x\left(\phi \to \psi \right)\right)\to \left(\exists !x\psi \to \left(\exists x\phi \wedge {\exists }^{*}x\phi \right)\right)$$
4		df-eu	$$⊢\exists !x\phi \leftrightarrow \left(\exists x\phi \wedge {\exists }^{*}x\phi \right)$$
5	3 4	syl6ibr	$$⊢\left(\exists x\phi \wedge \forall x\left(\phi \to \psi \right)\right)\to \left(\exists !x\psi \to \exists !x\phi \right)$$