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Theorem nfopab2 4520
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 4512 . 2  |-  { <. x ,  y >.  |  ph }  =  { z  |  E. x E. y
( z  =  <. x ,  y >.  /\  ph ) }
2 nfe1 1789 . . . 4  |-  F/ y E. y ( z  =  <. x ,  y
>.  /\  ph )
32nfex 1895 . . 3  |-  F/ y E. x E. y
( z  =  <. x ,  y >.  /\  ph )
43nfab 2633 . 2  |-  F/_ y { z  |  E. x E. y ( z  =  <. x ,  y
>.  /\  ph ) }
51, 4nfcxfr 2627 1  |-  F/_ y { <. x ,  y
>.  |  ph }
Colors of variables: wff setvar class
Syntax hints:    /\ wa 369    = wceq 1379   E.wex 1596   {cab 2452   F/_wnfc 2615   <.cop 4039   {copab 4510
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445
This theorem depends on definitions:  df-bi 185  df-an 371  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-opab 4512
This theorem is referenced by:  opelopabsb  4763  ssopab2b  4780  dmopab  5219  rnopab  5253  funopab  5627  fvopab5  5980  0neqopab  6336  zfrep6  6763  opabdm  27295  opabrn  27296  fpwrelmap  27387  aomclem8  30941  areaquad  31119
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