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Theorem nfopab2 4443
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 4435 . 2  |-  { <. x ,  y >.  |  ph }  =  { z  |  E. x E. y
( z  =  <. x ,  y >.  /\  ph ) }
2 nfe1 1779 . . . 4  |-  F/ y E. y ( z  =  <. x ,  y
>.  /\  ph )
32nfex 1875 . . 3  |-  F/ y E. x E. y
( z  =  <. x ,  y >.  /\  ph )
43nfab 2614 . 2  |-  F/_ y { z  |  E. x E. y ( z  =  <. x ,  y
>.  /\  ph ) }
51, 4nfcxfr 2608 1  |-  F/_ y { <. x ,  y
>.  |  ph }
Colors of variables: wff setvar class
Syntax hints:    /\ wa 369    = wceq 1370   E.wex 1587   {cab 2435   F/_wnfc 2596   <.cop 3967   {copab 4433
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1709  ax-7 1729  ax-10 1776  ax-11 1781  ax-12 1793  ax-13 1944  ax-ext 2429
This theorem depends on definitions:  df-bi 185  df-an 371  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1702  df-clab 2436  df-cleq 2442  df-clel 2445  df-nfc 2598  df-opab 4435
This theorem is referenced by:  opelopabsb  4683  ssopab2b  4699  dmopab  5134  rnopab  5168  funopab  5535  fvopab5  5880  0neqopab  6216  zfrep6  6631  opabdm  26063  opabrn  26064  fpwrelmap  26153  aomclem8  29538  areaquad  29716
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