Theorem sbex

Description: Move existential quantifier in and out of substitution. (Contributed by NM, 27-Sep-2003)

	Ref	Expression
Assertion	sbex	$$⊢\left[z/y\right]\exists x\phi \leftrightarrow \exists x\left[z/y\right]\phi$$

Proof

Step	Hyp	Ref	Expression
1		sbn	$$⊢\left[z/y\right]\neg \forall x\neg \phi \leftrightarrow \neg \left[z/y\right]\forall x\neg \phi$$
2		sbn	$$⊢\left[z/y\right]\neg \phi \leftrightarrow \neg \left[z/y\right]\phi$$
3	2	sbalv	$$⊢\left[z/y\right]\forall x\neg \phi \leftrightarrow \forall x\neg \left[z/y\right]\phi$$
4	1 3	xchbinx	$$⊢\left[z/y\right]\neg \forall x\neg \phi \leftrightarrow \neg \forall x\neg \left[z/y\right]\phi$$
5		df-ex	$$⊢\exists x\phi \leftrightarrow \neg \forall x\neg \phi$$
6	5	sbbii	$$⊢\left[z/y\right]\exists x\phi \leftrightarrow \left[z/y\right]\neg \forall x\neg \phi$$
7		df-ex	$$⊢\exists x\left[z/y\right]\phi \leftrightarrow \neg \forall x\neg \left[z/y\right]\phi$$
8	4 6 7	3bitr4i	$$⊢\left[z/y\right]\exists x\phi \leftrightarrow \exists x\left[z/y\right]\phi$$