Theorem nfoprab 5550

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

Hypothesis

Ref	Expression
nfoprab.1	|- F/wph

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 5529	. 2 |- {<.<.x, y>., z>. | ph} = {v | E.xE.yE.z(v = <.<.x, y>., z>. /\ ph)}
2		nfv 1619	. . . . . . 7 |- F/w v = <.<.x, y>., z>.
3		nfoprab.1	. . . . . . 7 |- F/wph
4	2, 3	nfan 1824	. . . . . 6 |- F/w(v = <.<.x, y>., z>. /\ ph)
5	4	nfex 1843	. . . . 5 |- F/wE.z(v = <.<.x, y>., z>. /\ ph)
6	5	nfex 1843	. . . 4 |- F/wE.yE.z(v = <.<.x, y>., z>. /\ ph)
7	6	nfex 1843	. . 3 |- F/wE.xE.yE.z(v = <.<.x, y>., z>. /\ ph)
8	7	nfab 2494	. 2 |- F/_w{v | E.xE.yE.z(v = <.<.x, y>., z>. /\ ph)}
9	1, 8	nfcxfr 2487	1 |- F/_w{<.<.x, y>., z>. | ph}

Syntax hints:   /\ wa 358  E.wex 1541   F/wnf 1544   = wceq 1642  {cab 2339   F/_wnfc 2477  <.cop 4562  {coprab 5528

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334

This theorem depends on definitions:  df-bi 177  df-an 360  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2479  df-oprab 5529

This theorem is referenced by:  nfmpt2  5676