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Theorem nfoprab 6058
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.
StepHypRef Expression
1 df-oprab 6017 . 2  |-  { <. <.
x ,  y >. ,  z >.  |  ph }  =  { v  |  E. x E. y E. z ( v  = 
<. <. x ,  y
>. ,  z >.  /\ 
ph ) }
2 nfv 1626 . . . . . . 7  |-  F/ w  v  =  <. <. x ,  y >. ,  z
>.
3 nfoprab.1 . . . . . . 7  |-  F/ w ph
42, 3nfan 1836 . . . . . 6  |-  F/ w
( v  =  <. <.
x ,  y >. ,  z >.  /\  ph )
54nfex 1855 . . . . 5  |-  F/ w E. z ( v  = 
<. <. x ,  y
>. ,  z >.  /\ 
ph )
65nfex 1855 . . . 4  |-  F/ w E. y E. z ( v  =  <. <. x ,  y >. ,  z
>.  /\  ph )
76nfex 1855 . . 3  |-  F/ w E. x E. y E. z ( v  = 
<. <. x ,  y
>. ,  z >.  /\ 
ph )
87nfab 2520 . 2  |-  F/_ w { v  |  E. x E. y E. z
( v  =  <. <.
x ,  y >. ,  z >.  /\  ph ) }
91, 8nfcxfr 2513 1  |-  F/_ w { <. <. x ,  y
>. ,  z >.  | 
ph }
Colors of variables: wff set class
Syntax hints:    /\ wa 359   E.wex 1547   F/wnf 1550    = wceq 1649   {cab 2366   F/_wnfc 2503   <.cop 3753   {coprab 6014
This theorem is referenced by:  nfmpt2  6074
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2361
This theorem depends on definitions:  df-bi 178  df-an 361  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-clab 2367  df-cleq 2373  df-clel 2376  df-nfc 2505  df-oprab 6017
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