Theorem nfopab 3825

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 3819	. 2 |- { <. x , y >. | ph } = { w | E. x E. y ( w = <. x , y >. /\ ph ) }
2		nfv 1421	. . . . . 6 |- F/ z w = <. x , y >.
3		nfopab.1	. . . . . 6 |- F/ z ph
4	2, 3	nfan 1457	. . . . 5 |- F/ z ( w = <. x , y >. /\ ph )
5	4	nfex 1528	. . . 4 |- F/ z E. y ( w = <. x , y >. /\ ph )
6	5	nfex 1528	. . 3 |- F/ z E. x E. y ( w = <. x , y >. /\ ph )
7	6	nfab 2182	. 2 |- F/_ z { w | E. x E. y ( w = <. x , y >. /\ ph ) }
8	1, 7	nfcxfr 2175	1 |- F/_ z { <. x , y >. | ph }

Syntax hints:   /\ wa 97   = wceq 1243   F/wnf 1349   E.wex 1381   {cab 2026   F/_wnfc 2165   <.cop 3378   {copab 3817

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022

This theorem depends on definitions:  df-bi 110  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-opab 3819

This theorem is referenced by:  csbopabg  3835  nfmpt  3849  nfxp  4371  nfco  4501  nfcnv  4514