Theorem hbabg

Description: Bound-variable hypothesis builder for a class abstraction. Usage of this theorem is discouraged because it depends on ax-13 . See hbab for a version with more disjoint variable conditions, but not requiring ax-13 . (Contributed by NM, 1-Mar-1995) (New usage is discouraged.)

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

Proof

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