Theorem wl-sb8eft

Description: Substitution of variable in existentialal quantifier. Closed form of sb8ef . (Contributed by Wolf Lammen, 27-Apr-2025)

	Ref	Expression
Assertion	wl-sb8eft	$$⊢\forall xℲy\phi \to \left(\exists x\phi \leftrightarrow \exists y\left[y/x\right]\phi \right)$$

Proof

Step	Hyp	Ref	Expression
1		nfnt	$$⊢Ⅎy\phi \to Ⅎy\neg \phi$$
2	1	alimi	$$⊢\forall xℲy\phi \to \forall xℲy\neg \phi$$
3		wl-sb8ft	$$⊢\forall xℲy\neg \phi \to \left(\forall x\neg \phi \leftrightarrow \forall y\left[y/x\right]\neg \phi \right)$$
4	2 3	syl	$$⊢\forall xℲy\phi \to \left(\forall x\neg \phi \leftrightarrow \forall y\left[y/x\right]\neg \phi \right)$$
5		alnex	$$⊢\forall x\neg \phi \leftrightarrow \neg \exists x\phi$$
6		sbn	$$⊢\left[y/x\right]\neg \phi \leftrightarrow \neg \left[y/x\right]\phi$$
7	6	albii	$$⊢\forall y\left[y/x\right]\neg \phi \leftrightarrow \forall y\neg \left[y/x\right]\phi$$
8		alnex	$$⊢\forall y\neg \left[y/x\right]\phi \leftrightarrow \neg \exists y\left[y/x\right]\phi$$
9	7 8	bitri	$$⊢\forall y\left[y/x\right]\neg \phi \leftrightarrow \neg \exists y\left[y/x\right]\phi$$
10	4 5 9	3bitr3g	$$⊢\forall xℲy\phi \to \left(\neg \exists x\phi \leftrightarrow \neg \exists y\left[y/x\right]\phi \right)$$
11	10	con4bid	$$⊢\forall xℲy\phi \to \left(\exists x\phi \leftrightarrow \exists y\left[y/x\right]\phi \right)$$