Theorem nfop

Description: Bound-variable hypothesis builder for ordered pairs. (Contributed by NM, 14-Nov-1995)

	Ref	Expression
Hypotheses	nfop.1	$$⊢\underset{\_}{Ⅎ}xA$$
	nfop.2	$$⊢\underset{\_}{Ⅎ}xB$$
Assertion	nfop	$$⊢\underset{\_}{Ⅎ}x⟨A,B⟩$$

Proof

Step	Hyp	Ref	Expression
1		nfop.1	$$⊢\underset{\_}{Ⅎ}xA$$
2		nfop.2	$$⊢\underset{\_}{Ⅎ}xB$$
3		dfopif	$$⊢⟨A,B⟩=if\left(\left(A\in \mathrm{V}\wedge B\in \mathrm{V}\right),\left\{\left\{A\right\},\left\{A,B\right\}\right\},\varnothing \right)$$
4	1	nfel1	$$⊢ℲxA\in \mathrm{V}$$
5	2	nfel1	$$⊢ℲxB\in \mathrm{V}$$
6	4 5	nfan	$$⊢Ⅎx\left(A\in \mathrm{V}\wedge B\in \mathrm{V}\right)$$
7	1	nfsn	$$⊢\underset{\_}{Ⅎ}x\left\{A\right\}$$
8	1 2	nfpr	$$⊢\underset{\_}{Ⅎ}x\left\{A,B\right\}$$
9	7 8	nfpr	$$⊢\underset{\_}{Ⅎ}x\left\{\left\{A\right\},\left\{A,B\right\}\right\}$$
10		nfcv	$$⊢\underset{\_}{Ⅎ}x\varnothing$$
11	6 9 10	nfif	$$⊢\underset{\_}{Ⅎ}xif\left(\left(A\in \mathrm{V}\wedge B\in \mathrm{V}\right),\left\{\left\{A\right\},\left\{A,B\right\}\right\},\varnothing \right)$$
12	3 11	nfcxfr	$$⊢\underset{\_}{Ⅎ}x⟨A,B⟩$$