Theorem nfixp1

Description: The index variable in an indexed Cartesian product is not free. (Contributed by Jeff Madsen, 19-Jun-2011) (Revised by Mario Carneiro, 15-Oct-2016)

	Ref	Expression
Assertion	nfixp1	$$⊢\underset{\_}{Ⅎ}x\underset{x\in A}{⨉}B$$

Proof

Step	Hyp	Ref	Expression
1		df-ixp	$$⊢\underset{x\in A}{⨉}B=\left\{y|\left(yFn\left\{x|x\in A\right\}\wedge \forall x\in Ay\left(x\right)\in B\right)\right\}$$
2		nfcv	$$⊢\underset{\_}{Ⅎ}xy$$
3		nfab1	$$⊢\underset{\_}{Ⅎ}x\left\{x|x\in A\right\}$$
4	2 3	nffn	$$⊢ℲxyFn\left\{x|x\in A\right\}$$
5		nfra1	$$⊢Ⅎx\forall x\in Ay\left(x\right)\in B$$
6	4 5	nfan	$$⊢Ⅎx\left(yFn\left\{x|x\in A\right\}\wedge \forall x\in Ay\left(x\right)\in B\right)$$
7	6	nfab	$$⊢\underset{\_}{Ⅎ}x\left\{y|\left(yFn\left\{x|x\in A\right\}\wedge \forall x\in Ay\left(x\right)\in B\right)\right\}$$
8	1 7	nfcxfr	$$⊢\underset{\_}{Ⅎ}x\underset{x\in A}{⨉}B$$