Theorem sb8mo

Description: Variable substitution for the at-most-one quantifier. Usage of this theorem is discouraged because it depends on ax-13 . (Contributed by Alexander van der Vekens, 17-Jun-2017) (New usage is discouraged.)

	Ref	Expression
Hypothesis	sb8eu.1	$$⊢Ⅎy\phi$$
Assertion	sb8mo	$$⊢{\exists }^{*}x\phi \leftrightarrow {\exists }^{*}y\left[y/x\right]\phi$$

Proof

Step	Hyp	Ref	Expression
1		sb8eu.1	$$⊢Ⅎy\phi$$
2	1	sb8e	$$⊢\exists x\phi \leftrightarrow \exists y\left[y/x\right]\phi$$
3	1	sb8eu	$$⊢\exists !x\phi \leftrightarrow \exists !y\left[y/x\right]\phi$$
4	2 3	imbi12i	$$⊢\left(\exists x\phi \to \exists !x\phi \right)\leftrightarrow \left(\exists y\left[y/x\right]\phi \to \exists !y\left[y/x\right]\phi \right)$$
5		moeu	$$⊢{\exists }^{*}x\phi \leftrightarrow \left(\exists x\phi \to \exists !x\phi \right)$$
6		moeu	$$⊢{\exists }^{*}y\left[y/x\right]\phi \leftrightarrow \left(\exists y\left[y/x\right]\phi \to \exists !y\left[y/x\right]\phi \right)$$
7	4 5 6	3bitr4i	$$⊢{\exists }^{*}x\phi \leftrightarrow {\exists }^{*}y\left[y/x\right]\phi$$