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Theorem nfopab2 4434
Description: The second 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
nfopab2  |-  F/_ y { <. x ,  y
>.  |  ph }

Proof of Theorem nfopab2
Dummy variable  z is distinct from all other variables.
StepHypRef Expression
1 df-opab 4426 . 2  |-  { <. x ,  y >.  |  ph }  =  { z  |  E. x E. y
( z  =  <. x ,  y >.  /\  ph ) }
2 nfe1 1894 . . . 4  |-  F/ y E. y ( z  =  <. x ,  y
>.  /\  ph )
32nfex 2008 . . 3  |-  F/ y E. x E. y
( z  =  <. x ,  y >.  /\  ph )
43nfab 2573 . 2  |-  F/_ y { z  |  E. x E. y ( z  =  <. x ,  y
>.  /\  ph ) }
51, 4nfcxfr 2567 1  |-  F/_ y { <. x ,  y
>.  |  ph }
Colors of variables: wff setvar class
Syntax hints:    /\ wa 370    = wceq 1437   E.wex 1657   {cab 2414   F/_wnfc 2556   <.cop 3947   {copab 4424
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 2063  ax-ext 2408
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 2415  df-cleq 2421  df-clel 2424  df-nfc 2558  df-opab 4426
This theorem is referenced by:  opelopabsb  4673  ssopab2b  4690  dmopab  5007  rnopab  5041  funopab  5577  fvopab5  5933  0neqopab  6293  zfrep6  6719  opabdm  28165  opabrn  28166  fpwrelmap  28268  aomclem8  35832  areaquad  36014
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