Theorem mosub

Description: "At most one" remains true after substitution. (Contributed by NM, 9-Mar-1995)

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

Proof

Step	Hyp	Ref	Expression
1		mosub.1	$$⊢{\exists }^{*}x\phi$$
2		moeq	$$⊢{\exists }^{*}yy=A$$
3	1	ax-gen	$$⊢\forall y{\exists }^{*}x\phi$$
4		moexexvw	$$⊢\left({\exists }^{*}yy=A\wedge \forall y{\exists }^{*}x\phi \right)\to {\exists }^{*}x\exists y\left(y=A\wedge \phi \right)$$
5	2 3 4	mp2an	$$⊢{\exists }^{*}x\exists y\left(y=A\wedge \phi \right)$$