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Theorem nfopab2 4088
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 4080 . 2  |-  { <. x ,  y >.  |  ph }  =  { z  |  E. x E. y
( z  =  <. x ,  y >.  /\  ph ) }
2 nfe1 1507 . . . 4  |-  F/ y E. y ( z  =  <. x ,  y
>.  /\  ph )
32nfex 1648 . . 3  |-  F/ y E. x E. y
( z  =  <. x ,  y >.  /\  ph )
43nfab 2337 . 2  |-  F/_ y { z  |  E. x E. y ( z  =  <. x ,  y
>.  /\  ph ) }
51, 4nfcxfr 2329 1  |-  F/_ y { <. x ,  y
>.  |  ph }
Colors of variables: wff set class
Syntax hints:    /\ wa 104    = wceq 1364   E.wex 1503   {cab 2175   F/_wnfc 2319   <.cop 3610   {copab 4078
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2171
This theorem depends on definitions:  df-bi 117  df-nf 1472  df-sb 1774  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-opab 4080
This theorem is referenced by:  opelopabsb  4275  ssopab2b  4291  dmopab  4853  rnopab  4889  funopab  5267  0neqopab  5937
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