Theorem nfopab1

Description: The first abstraction variable in an ordered-pair class abstraction is effectively not free. (Contributed by NM, 16-May-1995) (Revised by Mario Carneiro, 14-Oct-2016)

	Ref	Expression
Assertion	nfopab1	$$⊢\underset{\_}{Ⅎ}x\left\{⟨x,y⟩|\phi \right\}$$

Proof

Step	Hyp	Ref	Expression
1		df-opab	$$⊢\left\{⟨x,y⟩|\phi \right\}=\left\{z|\exists x\exists y\left(z=⟨x,y⟩\wedge \phi \right)\right\}$$
2		nfe1	$$⊢Ⅎx\exists x\exists y\left(z=⟨x,y⟩\wedge \phi \right)$$
3	2	nfab	$$⊢\underset{\_}{Ⅎ}x\left\{z|\exists x\exists y\left(z=⟨x,y⟩\wedge \phi \right)\right\}$$
4	1 3	nfcxfr	$$⊢\underset{\_}{Ⅎ}x\left\{⟨x,y⟩|\phi \right\}$$