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Theorem nfopab 4276
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 4270 . 2  |-  { <. x ,  y >.  |  ph }  =  { w  |  E. x E. y
( w  =  <. x ,  y >.  /\  ph ) }
2 nfv 1630 . . . . . 6  |-  F/ z  w  =  <. x ,  y >.
3 nfopab.1 . . . . . 6  |-  F/ z
ph
42, 3nfan 1847 . . . . 5  |-  F/ z ( w  =  <. x ,  y >.  /\  ph )
54nfex 1866 . . . 4  |-  F/ z E. y ( w  =  <. x ,  y
>.  /\  ph )
65nfex 1866 . . 3  |-  F/ z E. x E. y
( w  =  <. x ,  y >.  /\  ph )
76nfab 2578 . 2  |-  F/_ z { w  |  E. x E. y ( w  =  <. x ,  y
>.  /\  ph ) }
81, 7nfcxfr 2571 1  |-  F/_ z { <. x ,  y
>.  |  ph }
Colors of variables: wff set class
Syntax hints:    /\ wa 360   E.wex 1551   F/wnf 1554    = wceq 1653   {cab 2424   F/_wnfc 2561   <.cop 3819   {copab 4268
This theorem is referenced by:  csbopabg  4286  nfmpt  4300  nfxp  4907  nfco  5041  nfcnv  5054
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419
This theorem depends on definitions:  df-bi 179  df-an 362  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-opab 4270
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