Theorem nfopab

Description: Bound-variable hypothesis builder for class abstraction. (Contributed by NM, 1-Sep-1999) Remove disjoint variable conditions. (Revised by Andrew Salmon, 11-Jul-2011)

	Ref	Expression
Hypothesis	nfopab.1	$$⊢Ⅎz\phi$$
Assertion	nfopab	$$⊢\underset{\_}{Ⅎ}z\left\{⟨x,y⟩|\phi \right\}$$

Proof

Step	Hyp	Ref	Expression
1		nfopab.1	$$⊢Ⅎz\phi$$
2		df-opab	$$⊢\left\{⟨x,y⟩|\phi \right\}=\left\{w|\exists x\exists y\left(w=⟨x,y⟩\wedge \phi \right)\right\}$$
3		nfv	$$⊢Ⅎzw=⟨x,y⟩$$
4	3 1	nfan	$$⊢Ⅎz\left(w=⟨x,y⟩\wedge \phi \right)$$
5	4	nfex	$$⊢Ⅎz\exists y\left(w=⟨x,y⟩\wedge \phi \right)$$
6	5	nfex	$$⊢Ⅎz\exists x\exists y\left(w=⟨x,y⟩\wedge \phi \right)$$
7	6	nfab	$$⊢\underset{\_}{Ⅎ}z\left\{w|\exists x\exists y\left(w=⟨x,y⟩\wedge \phi \right)\right\}$$
8	2 7	nfcxfr	$$⊢\underset{\_}{Ⅎ}z\left\{⟨x,y⟩|\phi \right\}$$