MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  nfoprab3 Unicode version

Theorem nfoprab3 6065
Description: The abstraction variables in an operation class abstraction are not free. (Contributed by NM, 22-Aug-2013.)
Assertion
Ref Expression
nfoprab3  |-  F/_ z { <. <. x ,  y
>. ,  z >.  | 
ph }

Proof of Theorem nfoprab3
Dummy variable  w is distinct from all other variables.
StepHypRef Expression
1 df-oprab 6025 . 2  |-  { <. <.
x ,  y >. ,  z >.  |  ph }  =  { w  |  E. x E. y E. z ( w  = 
<. <. x ,  y
>. ,  z >.  /\ 
ph ) }
2 nfe1 1739 . . . . 5  |-  F/ z E. z ( w  =  <. <. x ,  y
>. ,  z >.  /\ 
ph )
32nfex 1855 . . . 4  |-  F/ z E. y E. z
( w  =  <. <.
x ,  y >. ,  z >.  /\  ph )
43nfex 1855 . . 3  |-  F/ z E. x E. y E. z ( w  = 
<. <. x ,  y
>. ,  z >.  /\ 
ph )
54nfab 2528 . 2  |-  F/_ z { w  |  E. x E. y E. z
( w  =  <. <.
x ,  y >. ,  z >.  /\  ph ) }
61, 5nfcxfr 2521 1  |-  F/_ z { <. <. x ,  y
>. ,  z >.  | 
ph }
Colors of variables: wff set class
Syntax hints:    /\ wa 359   E.wex 1547    = wceq 1649   {cab 2374   F/_wnfc 2511   <.cop 3761   {coprab 6022
This theorem is referenced by:  ssoprab2b  6071  ov3  6150  tposoprab  6452
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 2369
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 2375  df-cleq 2381  df-clel 2384  df-nfc 2513  df-oprab 6025
  Copyright terms: Public domain W3C validator