Theorem nfopab2 4006

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.

Step	Hyp	Ref	Expression
1		df-opab 3998	. 2 |- { <. x , y >. | ph } = { z | E. x E. y ( z = <. x , y >. /\ ph ) }
2		nfe1 1473	. . . 4 |- F/ y E. y ( z = <. x , y >. /\ ph )
3	2	nfex 1617	. . 3 |- F/ y E. x E. y ( z = <. x , y >. /\ ph )
4	3	nfab 2287	. 2 |- F/_ y { z | E. x E. y ( z = <. x , y >. /\ ph ) }
5	1, 4	nfcxfr 2279	1 |- F/_ y { <. x , y >. | ph }

Syntax hints:   /\ wa 103   = wceq 1332   E.wex 1469   {cab 2126   F/_wnfc 2269   <.cop 3535   {copab 3996

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 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122

This theorem depends on definitions:  df-bi 116  df-nf 1438  df-sb 1737  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-opab 3998

This theorem is referenced by:  opelopabsb  4190  ssopab2b  4206  dmopab  4758  rnopab  4794  funopab  5166  0neqopab  5824