Theorem nfaba1

Description: Bound-variable hypothesis builder for a class abstraction. (Contributed by Mario Carneiro, 14-Oct-2016) Add disjoint variable condition to avoid ax-13 . See nfaba1g for a less restrictive version requiring more axioms. (Revised by GG, 20-Jan-2024) Avoid ax-12 . (Revised by SN, 14-May-2025)

	Ref	Expression
Assertion	nfaba1	$$⊢\underset{\_}{Ⅎ}x\left\{y|\forall x\phi \right\}$$

Proof

Step	Hyp	Ref	Expression
1		df-clab	$$⊢z\in \left\{y|\forall x\phi \right\}\leftrightarrow \left[z/y\right]\forall x\phi$$
2		sbal	$$⊢\left[z/y\right]\forall x\phi \leftrightarrow \forall x\left[z/y\right]\phi$$
3		nfa1	$$⊢Ⅎx\forall x\left[z/y\right]\phi$$
4	2 3	nfxfr	$$⊢Ⅎx\left[z/y\right]\forall x\phi$$
5	1 4	nfxfr	$$⊢Ⅎxz\in \left\{y|\forall x\phi \right\}$$
6	5	nfci	$$⊢\underset{\_}{Ⅎ}x\left\{y|\forall x\phi \right\}$$