Theorem alneu

Description: If a statement holds for all sets, there is not a unique set for which the statement holds. (Contributed by Alexander van der Vekens, 28-Nov-2017)

	Ref	Expression
Assertion	alneu	$$⊢\forall x\phi \to \neg \exists !x\phi$$

Proof

Step	Hyp	Ref	Expression
1		eunex	$$⊢\exists !x\phi \to \exists x\neg \phi$$
2		exnal	$$⊢\exists x\neg \phi \leftrightarrow \neg \forall x\phi$$
3	1 2	sylib	$$⊢\exists !x\phi \to \neg \forall x\phi$$
4	3	con2i	$$⊢\forall x\phi \to \neg \exists !x\phi$$