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Theorem nfopab 3867
Description: Bound-variable hypothesis builder for class abstraction. (Contributed by NM, 1-Sep-1999.) (Unnecessary distinct variable restrictions were removed by Andrew Salmon, 11-Jul-2011.)
Hypothesis
Ref Expression
nfopab.1  |-  F/ z
ph
Assertion
Ref Expression
nfopab  |-  F/_ z { <. x ,  y
>.  |  ph }
Distinct variable groups:    x, z    y,
z
Allowed substitution hints:    ph( x, y, z)

Proof of Theorem nfopab
Dummy variable  w is distinct from all other variables.
StepHypRef Expression
1 df-opab 3861 . 2  |-  { <. x ,  y >.  |  ph }  =  { w  |  E. x E. y
( w  =  <. x ,  y >.  /\  ph ) }
2 nfv 1462 . . . . . 6  |-  F/ z  w  =  <. x ,  y >.
3 nfopab.1 . . . . . 6  |-  F/ z
ph
42, 3nfan 1498 . . . . 5  |-  F/ z ( w  =  <. x ,  y >.  /\  ph )
54nfex 1569 . . . 4  |-  F/ z E. y ( w  =  <. x ,  y
>.  /\  ph )
65nfex 1569 . . 3  |-  F/ z E. x E. y
( w  =  <. x ,  y >.  /\  ph )
76nfab 2227 . 2  |-  F/_ z { w  |  E. x E. y ( w  =  <. x ,  y
>.  /\  ph ) }
81, 7nfcxfr 2220 1  |-  F/_ z { <. x ,  y
>.  |  ph }
Colors of variables: wff set class
Syntax hints:    /\ wa 102    = wceq 1285   F/wnf 1390   E.wex 1422   {cab 2069   F/_wnfc 2210   <.cop 3420   {copab 3859
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 2065
This theorem depends on definitions:  df-bi 115  df-nf 1391  df-sb 1688  df-clab 2070  df-cleq 2076  df-clel 2079  df-nfc 2212  df-opab 3861
This theorem is referenced by:  csbopabg  3877  nfmpt  3891  nfxp  4418  nfco  4550  nfcnv  4563
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