Theorem nfopab 4050

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 4044	. 2 |- { <. x , y >. | ph } = { w | E. x E. y ( w = <. x , y >. /\ ph ) }
2		nfv 1516	. . . . . 6 |- F/ z w = <. x , y >.
3		nfopab.1	. . . . . 6 |- F/ z ph
4	2, 3	nfan 1553	. . . . 5 |- F/ z ( w = <. x , y >. /\ ph )
5	4	nfex 1625	. . . 4 |- F/ z E. y ( w = <. x , y >. /\ ph )
6	5	nfex 1625	. . 3 |- F/ z E. x E. y ( w = <. x , y >. /\ ph )
7	6	nfab 2313	. 2 |- F/_ z { w | E. x E. y ( w = <. x , y >. /\ ph ) }
8	1, 7	nfcxfr 2305	1 |- F/_ z { <. x , y >. | ph }

Syntax hints:   /\ wa 103   = wceq 1343   F/wnf 1448   E.wex 1480   {cab 2151   F/_wnfc 2295   <.cop 3579   {copab 4042

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-ext 2147

This theorem depends on definitions:  df-bi 116  df-nf 1449  df-sb 1751  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-opab 4044

This theorem is referenced by:  csbopabg  4060  nfmpt  4074  nfxp  4631  nfco  4769  nfcnv  4783  nfofr  6056