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Theorem nfopab1 3855
Description: The first abstraction variable in an ordered-pair class abstraction (class builder) is effectively not free. (Contributed by NM, 16-May-1995.) (Revised by Mario Carneiro, 14-Oct-2016.)
Assertion
Ref Expression
nfopab1  |-  F/_ x { <. x ,  y
>.  |  ph }

Proof of Theorem nfopab1
Dummy variable  z is distinct from all other variables.
StepHypRef Expression
1 df-opab 3848 . 2  |-  { <. x ,  y >.  |  ph }  =  { z  |  E. x E. y
( z  =  <. x ,  y >.  /\  ph ) }
2 nfe1 1426 . . 3  |-  F/ x E. x E. y ( z  =  <. x ,  y >.  /\  ph )
32nfab 2224 . 2  |-  F/_ x { z  |  E. x E. y ( z  =  <. x ,  y
>.  /\  ph ) }
41, 3nfcxfr 2217 1  |-  F/_ x { <. x ,  y
>.  |  ph }
Colors of variables: wff set class
Syntax hints:    /\ wa 102    = wceq 1285   E.wex 1422   {cab 2068   F/_wnfc 2207   <.cop 3409   {copab 3846
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 2064
This theorem depends on definitions:  df-bi 115  df-nf 1391  df-sb 1687  df-clab 2069  df-cleq 2075  df-clel 2078  df-nfc 2209  df-opab 3848
This theorem is referenced by:  nfmpt1  3879  opelopabsb  4023  ssopab2b  4039  dmopab  4574  rnopab  4609  funopab  4965  0neqopab  5581
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