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Theorem nfopab2 4057
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 4049 . 2  |-  { <. x ,  y >.  |  ph }  =  { z  |  E. x E. y
( z  =  <. x ,  y >.  /\  ph ) }
2 nfe1 1489 . . . 4  |-  F/ y E. y ( z  =  <. x ,  y
>.  /\  ph )
32nfex 1630 . . 3  |-  F/ y E. x E. y
( z  =  <. x ,  y >.  /\  ph )
43nfab 2317 . 2  |-  F/_ y { z  |  E. x E. y ( z  =  <. x ,  y
>.  /\  ph ) }
51, 4nfcxfr 2309 1  |-  F/_ y { <. x ,  y
>.  |  ph }
Colors of variables: wff set class
Syntax hints:    /\ wa 103    = wceq 1348   E.wex 1485   {cab 2156   F/_wnfc 2299   <.cop 3584   {copab 4047
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-ext 2152
This theorem depends on definitions:  df-bi 116  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-opab 4049
This theorem is referenced by:  opelopabsb  4243  ssopab2b  4259  dmopab  4820  rnopab  4856  funopab  5231  0neqopab  5895
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