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Theorem nfopab1 4305
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 4298 . 2  |-  { <. x ,  y >.  |  ph }  =  { z  |  E. x E. y
( z  =  <. x ,  y >.  /\  ph ) }
2 nfe1 1750 . . 3  |-  F/ x E. x E. y ( z  =  <. x ,  y >.  /\  ph )
32nfab 2583 . 2  |-  F/_ x { z  |  E. x E. y ( z  =  <. x ,  y
>.  /\  ph ) }
41, 3nfcxfr 2576 1  |-  F/_ x { <. x ,  y
>.  |  ph }
Colors of variables: wff set class
Syntax hints:    /\ wa 360   E.wex 1551    = wceq 1654   {cab 2429   F/_wnfc 2566   <.cop 3846   {copab 4296
This theorem is referenced by:  nfmpt1  4329  opelopabsb  4500  ssopab2b  4516  dmopab  5115  rnopab  5150  funopab  5521  zfrep6  6004  0neqopab  6155  fvopab5  6570  aomclem8  27248
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 1628  ax-9 1669  ax-8 1690  ax-6 1747  ax-7 1752  ax-11 1764  ax-12 1954  ax-ext 2424
This theorem depends on definitions:  df-bi 179  df-an 362  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1661  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2568  df-opab 4298
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