Theorem nfopab 4162

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.)

Hypothesis

Ref	Expression
nfopab.1	|- F/ z ph

Assertion

Ref	Expression
nfopab	|- F/_ z { <. x , y >. | ph }

Distinct variable groups:   x, z   y, z

Allowed substitution hints:   ph( x, y, z)

Proof of Theorem nfopab

Dummy variable w is distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-opab 4156	. 2 |- { <. x , y >. | ph } = { w | E. x E. y ( w = <. x , y >. /\ ph ) }
2		nfv 1577	. . . . . 6 |- F/ z w = <. x , y >.
3		nfopab.1	. . . . . 6 |- F/ z ph
4	2, 3	nfan 1614	. . . . 5 |- F/ z ( w = <. x , y >. /\ ph )
5	4	nfex 1686	. . . 4 |- F/ z E. y ( w = <. x , y >. /\ ph )
6	5	nfex 1686	. . 3 |- F/ z E. x E. y ( w = <. x , y >. /\ ph )
7	6	nfab 2380	. 2 |- F/_ z { w | E. x E. y ( w = <. x , y >. /\ ph ) }
8	1, 7	nfcxfr 2372	1 |- F/_ z { <. x , y >. | ph }

Syntax hints:   /\ wa 104   = wceq 1398   F/wnf 1509   E.wex 1541   {cab 2217   F/_wnfc 2362   <.cop 3676   {copab 4154

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-ext 2213

This theorem depends on definitions:  df-bi 117  df-nf 1510  df-sb 1811  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-opab 4156

This theorem is referenced by:  csbopabg  4172  nfmpt  4186  nfxp  4758  nfco  4901  nfcnv  4915  nfofr  6251