Theorem pm11.58

Description: Theorem *11.58 in WhiteheadRussell p. 165. (Contributed by Andrew Salmon, 24-May-2011)

	Ref	Expression
Assertion	pm11.58	$$⊢\exists x\phi \leftrightarrow \exists x\exists y\left(\phi \wedge \left[y/x\right]\phi \right)$$

Proof

Step	Hyp	Ref	Expression
1		19.8a	$$⊢\phi \to \exists x\phi$$
2		nfv	$$⊢Ⅎy\phi$$
3	2	sb8e	$$⊢\exists x\phi \leftrightarrow \exists y\left[y/x\right]\phi$$
4	1 3	sylib	$$⊢\phi \to \exists y\left[y/x\right]\phi$$
5	4	pm4.71i	$$⊢\phi \leftrightarrow \left(\phi \wedge \exists y\left[y/x\right]\phi \right)$$
6		19.42v	$$⊢\exists y\left(\phi \wedge \left[y/x\right]\phi \right)\leftrightarrow \left(\phi \wedge \exists y\left[y/x\right]\phi \right)$$
7	5 6	bitr4i	$$⊢\phi \leftrightarrow \exists y\left(\phi \wedge \left[y/x\right]\phi \right)$$
8	7	exbii	$$⊢\exists x\phi \leftrightarrow \exists x\exists y\left(\phi \wedge \left[y/x\right]\phi \right)$$