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Theorem nfoprab 5755
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 5710 . 2  |-  { <. <.
x ,  y >. ,  z >.  |  ph }  =  { v  |  E. x E. y E. z ( v  = 
<. <. x ,  y
>. ,  z >.  /\ 
ph ) }
2 nfv 1476 . . . . . . 7  |-  F/ w  v  =  <. <. x ,  y >. ,  z
>.
3 nfoprab.1 . . . . . . 7  |-  F/ w ph
42, 3nfan 1512 . . . . . 6  |-  F/ w
( v  =  <. <.
x ,  y >. ,  z >.  /\  ph )
54nfex 1584 . . . . 5  |-  F/ w E. z ( v  = 
<. <. x ,  y
>. ,  z >.  /\ 
ph )
65nfex 1584 . . . 4  |-  F/ w E. y E. z ( v  =  <. <. x ,  y >. ,  z
>.  /\  ph )
76nfex 1584 . . 3  |-  F/ w E. x E. y E. z ( v  = 
<. <. x ,  y
>. ,  z >.  /\ 
ph )
87nfab 2245 . 2  |-  F/_ w { v  |  E. x E. y E. z
( v  =  <. <.
x ,  y >. ,  z >.  /\  ph ) }
91, 8nfcxfr 2237 1  |-  F/_ w { <. <. x ,  y
>. ,  z >.  | 
ph }
Colors of variables: wff set class
Syntax hints:    /\ wa 103    = wceq 1299   F/wnf 1404   E.wex 1436   {cab 2086   F/_wnfc 2227   <.cop 3477   {coprab 5707
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 671  ax-5 1391  ax-7 1392  ax-gen 1393  ax-ie1 1437  ax-ie2 1438  ax-8 1450  ax-10 1451  ax-11 1452  ax-i12 1453  ax-bndl 1454  ax-4 1455  ax-17 1474  ax-i9 1478  ax-ial 1482  ax-i5r 1483  ax-ext 2082
This theorem depends on definitions:  df-bi 116  df-nf 1405  df-sb 1704  df-clab 2087  df-cleq 2093  df-clel 2096  df-nfc 2229  df-oprab 5710
This theorem is referenced by:  nfmpo  5772
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