Theorem eliminable-abelv

Description: A theorem used to prove the base case of the Eliminability Theorem (see section comment): abstraction belongs to variable. (Contributed by BJ, 30-Apr-2024) (Proof modification is discouraged.) (New usage is discouraged.)

	Ref	Expression
Assertion	eliminable-abelv	$$⊢\left\{x|\phi \right\}\in y\leftrightarrow \exists z\left(\forall t\left(t\in z\leftrightarrow \left[t/x\right]\phi \right)\wedge z\in y\right)$$

Proof

Step	Hyp	Ref	Expression
1		dfclel	$$⊢\left\{x|\phi \right\}\in y\leftrightarrow \exists z\left(z=\left\{x|\phi \right\}\wedge z\in y\right)$$
2		eliminable-veqab	$$⊢z=\left\{x|\phi \right\}\leftrightarrow \forall t\left(t\in z\leftrightarrow \left[t/x\right]\phi \right)$$
3	2	anbi1i	$$⊢\left(z=\left\{x|\phi \right\}\wedge z\in y\right)\leftrightarrow \left(\forall t\left(t\in z\leftrightarrow \left[t/x\right]\phi \right)\wedge z\in y\right)$$
4	3	exbii	$$⊢\exists z\left(z=\left\{x|\phi \right\}\wedge z\in y\right)\leftrightarrow \exists z\left(\forall t\left(t\in z\leftrightarrow \left[t/x\right]\phi \right)\wedge z\in y\right)$$
5	1 4	bitri	$$⊢\left\{x|\phi \right\}\in y\leftrightarrow \exists z\left(\forall t\left(t\in z\leftrightarrow \left[t/x\right]\phi \right)\wedge z\in y\right)$$