Theorem nfoprab 5894

Description: Bound-variable hypothesis builder for an operation class abstraction. (Contributed by NM, 22-Aug-2013.)

Hypothesis

Ref	Expression
nfoprab.1	|- F/ w ph

Assertion

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

Distinct variable groups:   x, w   y, w   z, w

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

Proof of Theorem nfoprab

Dummy variable v is distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-oprab 5846	. 2 |- { <. <. x , y >. , z >. | ph } = { v | E. x E. y E. z ( v = <. <. x , y >. , z >. /\ ph ) }
2		nfv 1516	. . . . . . 7 |- F/ w v = <. <. x , y >. , z >.
3		nfoprab.1	. . . . . . 7 |- F/ w ph
4	2, 3	nfan 1553	. . . . . 6 |- F/ w ( v = <. <. x , y >. , z >. /\ ph )
5	4	nfex 1625	. . . . 5 |- F/ w E. z ( v = <. <. x , y >. , z >. /\ ph )
6	5	nfex 1625	. . . 4 |- F/ w E. y E. z ( v = <. <. x , y >. , z >. /\ ph )
7	6	nfex 1625	. . 3 |- F/ w E. x E. y E. z ( v = <. <. x , y >. , z >. /\ ph )
8	7	nfab 2313	. 2 |- F/_ w { v | E. x E. y E. z ( v = <. <. x , y >. , z >. /\ ph ) }
9	1, 8	nfcxfr 2305	1 |- F/_ w { <. <. x , y >. , z >. | ph }

Syntax hints:   /\ wa 103   = wceq 1343   F/wnf 1448   E.wex 1480   {cab 2151   F/_wnfc 2295   <.cop 3579   {coprab 5843

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-oprab 5846

This theorem is referenced by:  nfmpo  5911