Theorem eliminable-abelab

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

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

Proof

Step	Hyp	Ref	Expression
1		dfclel	$$⊢\left\{x|\phi \right\}\in \left\{y|\psi \right\}\leftrightarrow \exists z\left(z=\left\{x|\phi \right\}\wedge z\in \left\{y|\psi \right\}\right)$$
2		eliminable-veqab	$$⊢z=\left\{x|\phi \right\}\leftrightarrow \forall t\left(t\in z\leftrightarrow \left[t/x\right]\phi \right)$$
3		eliminable-velab	$$⊢z\in \left\{y|\psi \right\}\leftrightarrow \left[z/y\right]\psi$$
4	2 3	anbi12i	$$⊢\left(z=\left\{x|\phi \right\}\wedge z\in \left\{y|\psi \right\}\right)\leftrightarrow \left(\forall t\left(t\in z\leftrightarrow \left[t/x\right]\phi \right)\wedge \left[z/y\right]\psi \right)$$
5	4	exbii	$$⊢\exists z\left(z=\left\{x|\phi \right\}\wedge z\in \left\{y|\psi \right\}\right)\leftrightarrow \exists z\left(\forall t\left(t\in z\leftrightarrow \left[t/x\right]\phi \right)\wedge \left[z/y\right]\psi \right)$$
6	1 5	bitri	$$⊢\left\{x|\phi \right\}\in \left\{y|\psi \right\}\leftrightarrow \exists z\left(\forall t\left(t\in z\leftrightarrow \left[t/x\right]\phi \right)\wedge \left[z/y\right]\psi \right)$$