Theorem nfopab 4083

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 4077	. 2 |- { <. x , y >. | ph } = { w | E. x E. y ( w = <. x , y >. /\ ph ) }
2		nfv 1538	. . . . . 6 |- F/ z w = <. x , y >.
3		nfopab.1	. . . . . 6 |- F/ z ph
4	2, 3	nfan 1575	. . . . 5 |- F/ z ( w = <. x , y >. /\ ph )
5	4	nfex 1647	. . . 4 |- F/ z E. y ( w = <. x , y >. /\ ph )
6	5	nfex 1647	. . 3 |- F/ z E. x E. y ( w = <. x , y >. /\ ph )
7	6	nfab 2334	. 2 |- F/_ z { w | E. x E. y ( w = <. x , y >. /\ ph ) }
8	1, 7	nfcxfr 2326	1 |- F/_ z { <. x , y >. | ph }

Syntax hints:   /\ wa 104   = wceq 1363   F/wnf 1470   E.wex 1502   {cab 2173   F/_wnfc 2316   <.cop 3607   {copab 4075

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 710  ax-5 1457  ax-7 1458  ax-gen 1459  ax-ie1 1503  ax-ie2 1504  ax-8 1514  ax-10 1515  ax-11 1516  ax-i12 1517  ax-bndl 1519  ax-4 1520  ax-17 1536  ax-i9 1540  ax-ial 1544  ax-i5r 1545  ax-ext 2169

This theorem depends on definitions:  df-bi 117  df-nf 1471  df-sb 1773  df-clab 2174  df-cleq 2180  df-clel 2183  df-nfc 2318  df-opab 4077

This theorem is referenced by:  csbopabg  4093  nfmpt  4107  nfxp  4665  nfco  4804  nfcnv  4818  nfofr  6102