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Theorem nfopab 4186
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 4180 . 2  |-  { <. x ,  y >.  |  ph }  =  { w  |  E. x E. y
( w  =  <. x ,  y >.  /\  ph ) }
2 nfv 1624 . . . . . 6  |-  F/ z  w  =  <. x ,  y >.
3 nfopab.1 . . . . . 6  |-  F/ z
ph
42, 3nfan 1834 . . . . 5  |-  F/ z ( w  =  <. x ,  y >.  /\  ph )
54nfex 1853 . . . 4  |-  F/ z E. y ( w  =  <. x ,  y
>.  /\  ph )
65nfex 1853 . . 3  |-  F/ z E. x E. y
( w  =  <. x ,  y >.  /\  ph )
76nfab 2506 . 2  |-  F/_ z { w  |  E. x E. y ( w  =  <. x ,  y
>.  /\  ph ) }
81, 7nfcxfr 2499 1  |-  F/_ z { <. x ,  y
>.  |  ph }
Colors of variables: wff set class
Syntax hints:    /\ wa 358   E.wex 1546   F/wnf 1549    = wceq 1647   {cab 2352   F/_wnfc 2489   <.cop 3732   {copab 4178
This theorem is referenced by:  csbopabg  4196  nfmpt  4210  nfxp  4818  nfco  4952  nfcnv  4963  nfofr  6211
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1551  ax-5 1562  ax-17 1621  ax-9 1659  ax-8 1680  ax-6 1734  ax-7 1739  ax-11 1751  ax-12 1937  ax-ext 2347
This theorem depends on definitions:  df-bi 177  df-an 360  df-tru 1324  df-ex 1547  df-nf 1550  df-sb 1654  df-clab 2353  df-cleq 2359  df-clel 2362  df-nfc 2491  df-opab 4180
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