Theorem nfscott

Description: Bound-variable hypothesis builder for the Scott operation. (Contributed by Rohan Ridenour, 11-Aug-2023)

	Ref	Expression
Hypothesis	nfscott.1	$$⊢\underset{\_}{Ⅎ}xA$$
Assertion	nfscott	$$⊢\underset{\_}{Ⅎ}xScottA$$

Proof

Step	Hyp	Ref	Expression
1		nfscott.1	$$⊢\underset{\_}{Ⅎ}xA$$
2		df-scott	$$⊢ScottA=\left\{y\in A|\forall z\in A\mathrm{rank}\left(y\right)\subseteq \mathrm{rank}\left(z\right)\right\}$$
3		nfv	$$⊢Ⅎx\mathrm{rank}\left(y\right)\subseteq \mathrm{rank}\left(z\right)$$
4	1 3	nfralw	$$⊢Ⅎx\forall z\in A\mathrm{rank}\left(y\right)\subseteq \mathrm{rank}\left(z\right)$$
5	4 1	nfrabw	$$⊢\underset{\_}{Ⅎ}x\left\{y\in A|\forall z\in A\mathrm{rank}\left(y\right)\subseteq \mathrm{rank}\left(z\right)\right\}$$
6	2 5	nfcxfr	$$⊢\underset{\_}{Ⅎ}xScottA$$