Theorem nfoprab 6130

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 6079	. 2 |- { <. <. x , y >. , z >. | ph } = { v | E. x E. y E. z ( v = <. <. x , y >. , z >. /\ ph ) }
2		nfv 1581	. . . . . . 7 |- F/ w v = <. <. x , y >. , z >.
3		nfoprab.1	. . . . . . 7 |- F/ w ph
4	2, 3	nfan 1618	. . . . . 6 |- F/ w ( v = <. <. x , y >. , z >. /\ ph )
5	4	nfex 1690	. . . . 5 |- F/ w E. z ( v = <. <. x , y >. , z >. /\ ph )
6	5	nfex 1690	. . . 4 |- F/ w E. y E. z ( v = <. <. x , y >. , z >. /\ ph )
7	6	nfex 1690	. . 3 |- F/ w E. x E. y E. z ( v = <. <. x , y >. , z >. /\ ph )
8	7	nfab 2397	. 2 |- F/_ w { v | E. x E. y E. z ( v = <. <. x , y >. , z >. /\ ph ) }
9	1, 8	nfcxfr 2389	1 |- F/_ w { <. <. x , y >. , z >. | ph }

Syntax hints:   /\ wa 104   = wceq 1402   F/wnf 1513   E.wex 1545   {cab 2224   F/_wnfc 2379   <.cop 3708   {coprab 6076

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 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-ext 2220

This theorem depends on definitions:  df-bi 117  df-nf 1514  df-sb 1816  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-oprab 6079

This theorem is referenced by:  nfmpo  6147