Theorem sbmo

Description: Substitution into an at-most-one quantifier. (Contributed by Jeff Madsen, 2-Sep-2009)

	Ref	Expression
Assertion	sbmo	$$⊢\left[y/x\right]{\exists }^{*}z\phi \leftrightarrow {\exists }^{*}z\left[y/x\right]\phi$$

Proof

Step	Hyp	Ref	Expression
1		sbex	$$⊢\left[y/x\right]\exists w\forall z\left(\phi \to z=w\right)\leftrightarrow \exists w\left[y/x\right]\forall z\left(\phi \to z=w\right)$$
2		nfv	$$⊢Ⅎxz=w$$
3	2	sblim	$$⊢\left[y/x\right]\left(\phi \to z=w\right)\leftrightarrow \left(\left[y/x\right]\phi \to z=w\right)$$
4	3	sbalv	$$⊢\left[y/x\right]\forall z\left(\phi \to z=w\right)\leftrightarrow \forall z\left(\left[y/x\right]\phi \to z=w\right)$$
5	4	exbii	$$⊢\exists w\left[y/x\right]\forall z\left(\phi \to z=w\right)\leftrightarrow \exists w\forall z\left(\left[y/x\right]\phi \to z=w\right)$$
6	1 5	bitri	$$⊢\left[y/x\right]\exists w\forall z\left(\phi \to z=w\right)\leftrightarrow \exists w\forall z\left(\left[y/x\right]\phi \to z=w\right)$$
7		dfmo	$$⊢{\exists }^{*}z\phi \leftrightarrow \exists w\forall z\left(\phi \to z=w\right)$$
8	7	sbbii	$$⊢\left[y/x\right]{\exists }^{*}z\phi \leftrightarrow \left[y/x\right]\exists w\forall z\left(\phi \to z=w\right)$$
9		dfmo	$$⊢{\exists }^{*}z\left[y/x\right]\phi \leftrightarrow \exists w\forall z\left(\left[y/x\right]\phi \to z=w\right)$$
10	6 8 9	3bitr4i	$$⊢\left[y/x\right]{\exists }^{*}z\phi \leftrightarrow {\exists }^{*}z\left[y/x\right]\phi$$