Theorem mosubop

Description: "At most one" remains true inside ordered pair quantification. (Contributed by NM, 28-May-1995)

	Ref	Expression
Hypothesis	mosubop.1	$$⊢{\exists }^{*}x\phi$$
Assertion	mosubop	$$⊢{\exists }^{*}x\exists y\exists z\left(A=⟨y,z⟩\wedge \phi \right)$$

Proof

Step	Hyp	Ref	Expression
1		mosubop.1	$$⊢{\exists }^{*}x\phi$$
2	1	gen2	$$⊢\forall y\forall z{\exists }^{*}x\phi$$
3		mosubopt	$$⊢\forall y\forall z{\exists }^{*}x\phi \to {\exists }^{*}x\exists y\exists z\left(A=⟨y,z⟩\wedge \phi \right)$$
4	2 3	ax-mp	$$⊢{\exists }^{*}x\exists y\exists z\left(A=⟨y,z⟩\wedge \phi \right)$$