Theorem hbab

Description: Bound-variable hypothesis builder for a class abstraction. (Contributed by NM, 1-Mar-1995) Add disjoint variable condition to avoid ax-13 . See hbabg for a less restrictive version requiring more axioms. (Revised by Gino Giotto, 20-Jan-2024)

	Ref	Expression
Hypothesis	hbab.1	$$⊢\phi \to \forall x\phi$$
Assertion	hbab	$$⊢z\in \left\{y|\phi \right\}\to \forall xz\in \left\{y|\phi \right\}$$

Proof

Step	Hyp	Ref	Expression
1		hbab.1	$$⊢\phi \to \forall x\phi$$
2		df-clab	$$⊢z\in \left\{y|\phi \right\}\leftrightarrow \left[z/y\right]\phi$$
3	1	hbsbw	$$⊢\left[z/y\right]\phi \to \forall x\left[z/y\right]\phi$$
4	2 3	hbxfrbi	$$⊢z\in \left\{y|\phi \right\}\to \forall xz\in \left\{y|\phi \right\}$$