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Theorem nfopab 3898
Description: Bound-variable hypothesis builder for class abstraction. (Contributed by NM, 1-Sep-1999.) Remove disjoint variable conditions. (Revised 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 3892 . 2  |-  { <. x ,  y >.  |  ph }  =  { w  |  E. x E. y
( w  =  <. x ,  y >.  /\  ph ) }
2 nfv 1466 . . . . . 6  |-  F/ z  w  =  <. x ,  y >.
3 nfopab.1 . . . . . 6  |-  F/ z
ph
42, 3nfan 1502 . . . . 5  |-  F/ z ( w  =  <. x ,  y >.  /\  ph )
54nfex 1573 . . . 4  |-  F/ z E. y ( w  =  <. x ,  y
>.  /\  ph )
65nfex 1573 . . 3  |-  F/ z E. x E. y
( w  =  <. x ,  y >.  /\  ph )
76nfab 2233 . 2  |-  F/_ z { w  |  E. x E. y ( w  =  <. x ,  y
>.  /\  ph ) }
81, 7nfcxfr 2225 1  |-  F/_ z { <. x ,  y
>.  |  ph }
Colors of variables: wff set class
Syntax hints:    /\ wa 102    = wceq 1289   F/wnf 1394   E.wex 1426   {cab 2074   F/_wnfc 2215   <.cop 3444   {copab 3890
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 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070
This theorem depends on definitions:  df-bi 115  df-nf 1395  df-sb 1693  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-opab 3892
This theorem is referenced by:  csbopabg  3908  nfmpt  3922  nfxp  4454  nfco  4589  nfcnv  4603  nfofr  5844
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