ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  nfoprab Unicode version

Theorem nfoprab 5588
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 5547 . 2  |-  { <. <.
x ,  y >. ,  z >.  |  ph }  =  { v  |  E. x E. y E. z ( v  = 
<. <. x ,  y
>. ,  z >.  /\ 
ph ) }
2 nfv 1462 . . . . . . 7  |-  F/ w  v  =  <. <. x ,  y >. ,  z
>.
3 nfoprab.1 . . . . . . 7  |-  F/ w ph
42, 3nfan 1498 . . . . . 6  |-  F/ w
( v  =  <. <.
x ,  y >. ,  z >.  /\  ph )
54nfex 1569 . . . . 5  |-  F/ w E. z ( v  = 
<. <. x ,  y
>. ,  z >.  /\ 
ph )
65nfex 1569 . . . 4  |-  F/ w E. y E. z ( v  =  <. <. x ,  y >. ,  z
>.  /\  ph )
76nfex 1569 . . 3  |-  F/ w E. x E. y E. z ( v  = 
<. <. x ,  y
>. ,  z >.  /\ 
ph )
87nfab 2224 . 2  |-  F/_ w { v  |  E. x E. y E. z
( v  =  <. <.
x ,  y >. ,  z >.  /\  ph ) }
91, 8nfcxfr 2217 1  |-  F/_ w { <. <. x ,  y
>. ,  z >.  | 
ph }
Colors of variables: wff set class
Syntax hints:    /\ wa 102    = wceq 1285   F/wnf 1390   E.wex 1422   {cab 2068   F/_wnfc 2207   <.cop 3409   {coprab 5544
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2064
This theorem depends on definitions:  df-bi 115  df-nf 1391  df-sb 1687  df-clab 2069  df-cleq 2075  df-clel 2078  df-nfc 2209  df-oprab 5547
This theorem is referenced by:  nfmpt2  5604
  Copyright terms: Public domain W3C validator