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Theorem nfopab 4084
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 4078 . 2  |-  { <. x ,  y >.  |  ph }  =  { w  |  E. x E. y
( w  =  <. x ,  y >.  /\  ph ) }
2 nfv 1605 . . . . . 6  |-  F/ z  w  =  <. x ,  y >.
3 nfopab.1 . . . . . 6  |-  F/ z
ph
42, 3nfan 1771 . . . . 5  |-  F/ z ( w  =  <. x ,  y >.  /\  ph )
54nfex 1767 . . . 4  |-  F/ z E. y ( w  =  <. x ,  y
>.  /\  ph )
65nfex 1767 . . 3  |-  F/ z E. x E. y
( w  =  <. x ,  y >.  /\  ph )
76nfab 2423 . 2  |-  F/_ z { w  |  E. x E. y ( w  =  <. x ,  y
>.  /\  ph ) }
81, 7nfcxfr 2416 1  |-  F/_ z { <. x ,  y
>.  |  ph }
Colors of variables: wff set class
Syntax hints:    /\ wa 358   E.wex 1528   F/wnf 1531    = wceq 1623   {cab 2269   F/_wnfc 2406   <.cop 3643   {copab 4076
This theorem is referenced by:  csbopabg  4094  nfmpt  4108  nfxp  4715  nfco  4849  nfcnv  4860  nfofr  6084
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264
This theorem depends on definitions:  df-bi 177  df-an 360  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-opab 4078
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