Theorem nfxp

Description: Bound-variable hypothesis builder for Cartesian product. (Contributed by NM, 15-Sep-2003) (Revised by Mario Carneiro, 15-Oct-2016)

	Ref	Expression
Hypotheses	nfxp.1	$$⊢\underset{\_}{Ⅎ}xA$$
	nfxp.2	$$⊢\underset{\_}{Ⅎ}xB$$
Assertion	nfxp	$$⊢\underset{\_}{Ⅎ}x\left(A\times B\right)$$

Proof

Step	Hyp	Ref	Expression
1		nfxp.1	$$⊢\underset{\_}{Ⅎ}xA$$
2		nfxp.2	$$⊢\underset{\_}{Ⅎ}xB$$
3		df-xp	$$⊢A\times B=\left\{⟨y,z⟩|\left(y\in A\wedge z\in B\right)\right\}$$
4	1	nfcri	$$⊢Ⅎxy\in A$$
5	2	nfcri	$$⊢Ⅎxz\in B$$
6	4 5	nfan	$$⊢Ⅎx\left(y\in A\wedge z\in B\right)$$
7	6	nfopab	$$⊢\underset{\_}{Ⅎ}x\left\{⟨y,z⟩|\left(y\in A\wedge z\in B\right)\right\}$$
8	3 7	nfcxfr	$$⊢\underset{\_}{Ⅎ}x\left(A\times B\right)$$