Theorem nfwrecs

Description: Bound-variable hypothesis builder for the well-ordered recursive function generator. (Contributed by Scott Fenton, 9-Jun-2018) (Proof shortened by Scott Fenton, 17-Nov-2024)

	Ref	Expression
Hypotheses	nfwrecs.1	$$⊢\underset{\_}{Ⅎ}xR$$
	nfwrecs.2	$$⊢\underset{\_}{Ⅎ}xA$$
	nfwrecs.3	$$⊢\underset{\_}{Ⅎ}xF$$
Assertion	nfwrecs	$$⊢\underset{\_}{Ⅎ}x\mathrm{wrecs}\left(R,A,F\right)$$

Proof

Step	Hyp	Ref	Expression
1		nfwrecs.1	$$⊢\underset{\_}{Ⅎ}xR$$
2		nfwrecs.2	$$⊢\underset{\_}{Ⅎ}xA$$
3		nfwrecs.3	$$⊢\underset{\_}{Ⅎ}xF$$
4		df-wrecs	$$⊢\mathrm{wrecs}\left(R,A,F\right)=\mathrm{frecs}\left(R,A,F\circ 2^{nd}\right)$$
5		nfcv	$$⊢\underset{\_}{Ⅎ}x{2}^{nd}$$
6	3 5	nfco	$$⊢\underset{\_}{Ⅎ}x\left(F\circ 2^{nd}\right)$$
7	1 2 6	nffrecs	$$⊢\underset{\_}{Ⅎ}x\mathrm{frecs}\left(R,A,F\circ 2^{nd}\right)$$
8	4 7	nfcxfr	$$⊢\underset{\_}{Ⅎ}x\mathrm{wrecs}\left(R,A,F\right)$$