Theorem exists2

Description: A condition implying that at least two things exist. (Contributed by NM, 10-Apr-2004) (Proof shortened by Andrew Salmon, 9-Jul-2011) Reduce axiom usage. (Revised by Wolf Lammen, 4-Mar-2023)

	Ref	Expression
Assertion	exists2	$$⊢\left(\exists x\phi \wedge \exists x\neg \phi \right)\to \neg \exists !xx=x$$

Proof

Step	Hyp	Ref	Expression
1		axc16nf	$$⊢\forall xx=y\to Ⅎx\phi$$
2	1	nfrd	$$⊢\forall xx=y\to \left(\exists x\phi \to \forall x\phi \right)$$
3	2	com12	$$⊢\exists x\phi \to \left(\forall xx=y\to \forall x\phi \right)$$
4		exists1	$$⊢\exists !xx=x\leftrightarrow \forall xx=y$$
5		alex	$$⊢\forall x\phi \leftrightarrow \neg \exists x\neg \phi$$
6	5	bicomi	$$⊢\neg \exists x\neg \phi \leftrightarrow \forall x\phi$$
7	3 4 6	3imtr4g	$$⊢\exists x\phi \to \left(\exists !xx=x\to \neg \exists x\neg \phi \right)$$
8	7	con2d	$$⊢\exists x\phi \to \left(\exists x\neg \phi \to \neg \exists !xx=x\right)$$
9	8	imp	$$⊢\left(\exists x\phi \wedge \exists x\neg \phi \right)\to \neg \exists !xx=x$$