Theorem nfoprab 5927

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 5879	. 2 |- { <. <. x , y >. , z >. | ph } = { v | E. x E. y E. z ( v = <. <. x , y >. , z >. /\ ph ) }
2		nfv 1528	. . . . . . 7 |- F/ w v = <. <. x , y >. , z >.
3		nfoprab.1	. . . . . . 7 |- F/ w ph
4	2, 3	nfan 1565	. . . . . 6 |- F/ w ( v = <. <. x , y >. , z >. /\ ph )
5	4	nfex 1637	. . . . 5 |- F/ w E. z ( v = <. <. x , y >. , z >. /\ ph )
6	5	nfex 1637	. . . 4 |- F/ w E. y E. z ( v = <. <. x , y >. , z >. /\ ph )
7	6	nfex 1637	. . 3 |- F/ w E. x E. y E. z ( v = <. <. x , y >. , z >. /\ ph )
8	7	nfab 2324	. 2 |- F/_ w { v | E. x E. y E. z ( v = <. <. x , y >. , z >. /\ ph ) }
9	1, 8	nfcxfr 2316	1 |- F/_ w { <. <. x , y >. , z >. | ph }

Syntax hints:   /\ wa 104   = wceq 1353   F/wnf 1460   E.wex 1492   {cab 2163   F/_wnfc 2306   <.cop 3596   {coprab 5876

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 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-ext 2159

This theorem depends on definitions:  df-bi 117  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-oprab 5879

This theorem is referenced by:  nfmpo  5944