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Theorem nfopab1 4490
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 4483 . 2  |-  { <. x ,  y >.  |  ph }  =  { z  |  E. x E. y
( z  =  <. x ,  y >.  /\  ph ) }
2 nfe1 1894 . . 3  |-  F/ x E. x E. y ( z  =  <. x ,  y >.  /\  ph )
32nfab 2584 . 2  |-  F/_ x { z  |  E. x E. y ( z  =  <. x ,  y
>.  /\  ph ) }
41, 3nfcxfr 2578 1  |-  F/_ x { <. x ,  y
>.  |  ph }
Colors of variables: wff setvar class
Syntax hints:    /\ wa 370    = wceq 1437   E.wex 1657   {cab 2407   F/_wnfc 2566   <.cop 4004   {copab 4481
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1663  ax-4 1676  ax-5 1752  ax-6 1798  ax-7 1843  ax-10 1891  ax-11 1896  ax-12 1909  ax-13 2057  ax-ext 2401
This theorem depends on definitions:  df-bi 188  df-an 372  df-tru 1440  df-ex 1658  df-nf 1662  df-sb 1791  df-clab 2408  df-cleq 2414  df-clel 2417  df-nfc 2568  df-opab 4483
This theorem is referenced by:  nfmpt1  4513  opelopabsb  4730  ssopab2b  4747  dmopab  5064  rnopab  5098  funopab  5634  fvopab5  5990  0neqopab  6350  zfrep6  6776  opabdm  28222  opabrn  28223  fpwrelmap  28325  aomclem8  35890
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