Theorem mo3

Description: Alternate definition of the at-most-one quantifier. Definition of BellMachover p. 460, except that definition has the side condition that y not occur in ph in place of our hypothesis. (Contributed by NM, 8-Mar-1995) (Proof shortened by Wolf Lammen, 18-Aug-2019) Remove dependency on ax-13 . (Revised by BJ and WL, 29-Jan-2023)

	Ref	Expression
Hypothesis	mo3.nf	$$⊢Ⅎy\phi$$
Assertion	mo3	$$⊢{\exists }^{*}x\phi \leftrightarrow \forall x\forall y\left(\left(\phi \wedge \left[y/x\right]\phi \right)\to x=y\right)$$

Proof

Step	Hyp	Ref	Expression
1		mo3.nf	$$⊢Ⅎy\phi$$
2		nfmo1	$$⊢Ⅎx{\exists }^{*}x\phi$$
3	1	nfmov	$$⊢Ⅎy{\exists }^{*}x\phi$$
4		df-mo	$$⊢{\exists }^{*}x\phi \leftrightarrow \exists z\forall x\left(\phi \to x=z\right)$$
5		sp	$$⊢\forall x\left(\phi \to x=z\right)\to \left(\phi \to x=z\right)$$
6		spsbim	$$⊢\forall x\left(\phi \to x=z\right)\to \left(\left[y/x\right]\phi \to \left[y/x\right]x=z\right)$$
7		equsb3	$$⊢\left[y/x\right]x=z\leftrightarrow y=z$$
8	6 7	imbitrdi	$$⊢\forall x\left(\phi \to x=z\right)\to \left(\left[y/x\right]\phi \to y=z\right)$$
9	5 8	anim12d	$$⊢\forall x\left(\phi \to x=z\right)\to \left(\left(\phi \wedge \left[y/x\right]\phi \right)\to \left(x=z\wedge y=z\right)\right)$$
10		equtr2	$$⊢\left(x=z\wedge y=z\right)\to x=y$$
11	9 10	syl6	$$⊢\forall x\left(\phi \to x=z\right)\to \left(\left(\phi \wedge \left[y/x\right]\phi \right)\to x=y\right)$$
12	11	exlimiv	$$⊢\exists z\forall x\left(\phi \to x=z\right)\to \left(\left(\phi \wedge \left[y/x\right]\phi \right)\to x=y\right)$$
13	4 12	sylbi	$$⊢{\exists }^{*}x\phi \to \left(\left(\phi \wedge \left[y/x\right]\phi \right)\to x=y\right)$$
14	3 13	alrimi	$$⊢{\exists }^{*}x\phi \to \forall y\left(\left(\phi \wedge \left[y/x\right]\phi \right)\to x=y\right)$$
15	2 14	alrimi	$$⊢{\exists }^{*}x\phi \to \forall x\forall y\left(\left(\phi \wedge \left[y/x\right]\phi \right)\to x=y\right)$$
16		nfs1v	$$⊢Ⅎx\left[y/x\right]\phi$$
17		pm3.21	$$⊢\left[y/x\right]\phi \to \left(\phi \to \left(\phi \wedge \left[y/x\right]\phi \right)\right)$$
18	17	imim1d	$$⊢\left[y/x\right]\phi \to \left(\left(\left(\phi \wedge \left[y/x\right]\phi \right)\to x=y\right)\to \left(\phi \to x=y\right)\right)$$
19	16 18	alimd	$$⊢\left[y/x\right]\phi \to \left(\forall x\left(\left(\phi \wedge \left[y/x\right]\phi \right)\to x=y\right)\to \forall x\left(\phi \to x=y\right)\right)$$
20	19	com12	$$⊢\forall x\left(\left(\phi \wedge \left[y/x\right]\phi \right)\to x=y\right)\to \left(\left[y/x\right]\phi \to \forall x\left(\phi \to x=y\right)\right)$$
21	20	aleximi	$$⊢\forall y\forall x\left(\left(\phi \wedge \left[y/x\right]\phi \right)\to x=y\right)\to \left(\exists y\left[y/x\right]\phi \to \exists y\forall x\left(\phi \to x=y\right)\right)$$
22	1	sb8ef	$$⊢\exists x\phi \leftrightarrow \exists y\left[y/x\right]\phi$$
23	1	mof	$$⊢{\exists }^{*}x\phi \leftrightarrow \exists y\forall x\left(\phi \to x=y\right)$$
24	21 22 23	3imtr4g	$$⊢\forall y\forall x\left(\left(\phi \wedge \left[y/x\right]\phi \right)\to x=y\right)\to \left(\exists x\phi \to {\exists }^{*}x\phi \right)$$
25		moabs	$$⊢{\exists }^{*}x\phi \leftrightarrow \left(\exists x\phi \to {\exists }^{*}x\phi \right)$$
26	24 25	sylibr	$$⊢\forall y\forall x\left(\left(\phi \wedge \left[y/x\right]\phi \right)\to x=y\right)\to {\exists }^{*}x\phi$$
27	26	alcoms	$$⊢\forall x\forall y\left(\left(\phi \wedge \left[y/x\right]\phi \right)\to x=y\right)\to {\exists }^{*}x\phi$$
28	15 27	impbii	$$⊢{\exists }^{*}x\phi \leftrightarrow \forall x\forall y\left(\left(\phi \wedge \left[y/x\right]\phi \right)\to x=y\right)$$