Theorem nfabd

Description: Bound-variable hypothesis builder for a class abstraction. Usage of this theorem is discouraged because it depends on ax-13 . Use the weaker nfabdw when possible. (Contributed by Mario Carneiro, 8-Oct-2016) Avoid ax-9 and ax-ext . (Revised by Wolf Lammen, 23-May-2023) (New usage is discouraged.)

	Ref	Expression
Hypotheses	nfabd.1	$$⊢Ⅎy\phi$$
	nfabd.2	$$⊢\phi \to Ⅎx\psi$$
Assertion	nfabd	$$⊢\phi \to \underset{\_}{Ⅎ}x\left\{y|\psi \right\}$$

Proof

Step	Hyp	Ref	Expression
1		nfabd.1	$$⊢Ⅎy\phi$$
2		nfabd.2	$$⊢\phi \to Ⅎx\psi$$
3		nfv	$$⊢Ⅎz\phi$$
4		df-clab	$$⊢z\in \left\{y|\psi \right\}\leftrightarrow \left[z/y\right]\psi$$
5	1 2	nfsbd	$$⊢\phi \to Ⅎx\left[z/y\right]\psi$$
6	4 5	nfxfrd	$$⊢\phi \to Ⅎxz\in \left\{y|\psi \right\}$$
7	3 6	nfcd	$$⊢\phi \to \underset{\_}{Ⅎ}x\left\{y|\psi \right\}$$