Theorem nfopab 3846

Description: Bound-variable hypothesis builder for class abstraction. (Contributed by NM, 1-Sep-1999.) (Unnecessary distinct variable restrictions were removed 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 3840	. 2 |- { <. x , y >. | ph } = { w | E. x E. y ( w = <. x , y >. /\ ph ) }
2		nfv 1461	. . . . . 6 |- F/ z w = <. x , y >.
3		nfopab.1	. . . . . 6 |- F/ z ph
4	2, 3	nfan 1497	. . . . 5 |- F/ z ( w = <. x , y >. /\ ph )
5	4	nfex 1568	. . . 4 |- F/ z E. y ( w = <. x , y >. /\ ph )
6	5	nfex 1568	. . 3 |- F/ z E. x E. y ( w = <. x , y >. /\ ph )
7	6	nfab 2223	. 2 |- F/_ z { w | E. x E. y ( w = <. x , y >. /\ ph ) }
8	1, 7	nfcxfr 2216	1 |- F/_ z { <. x , y >. | ph }

Syntax hints:   /\ wa 102   = wceq 1284   F/wnf 1389   E.wex 1421   {cab 2067   F/_wnfc 2206   <.cop 3401   {copab 3838

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-bndl 1439  ax-4 1440  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063

This theorem depends on definitions:  df-bi 115  df-nf 1390  df-sb 1686  df-clab 2068  df-cleq 2074  df-clel 2077  df-nfc 2208  df-opab 3840

This theorem is referenced by:  csbopabg  3856  nfmpt  3870  nfxp  4389  nfco  4519  nfcnv  4532