Theorem nfopab1 3997

Description: The first 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
nfopab1	|- F/_ x { <. x , y >. | ph }

Proof of Theorem nfopab1

Dummy variable z is distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-opab 3990	. 2 |- { <. x , y >. | ph } = { z | E. x E. y ( z = <. x , y >. /\ ph ) }
2		nfe1 1472	. . 3 |- F/ x E. x E. y ( z = <. x , y >. /\ ph )
3	2	nfab 2286	. 2 |- F/_ x { z | E. x E. y ( z = <. x , y >. /\ ph ) }
4	1, 3	nfcxfr 2278	1 |- F/_ x { <. x , y >. | ph }

Syntax hints:   /\ wa 103   = wceq 1331   E.wex 1468   {cab 2125   F/_wnfc 2268   <.cop 3530   {copab 3988

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 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121

This theorem depends on definitions:  df-bi 116  df-nf 1437  df-sb 1736  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-opab 3990

This theorem is referenced by:  nfmpt1  4021  opelopabsb  4182  ssopab2b  4198  dmopab  4750  rnopab  4786  funopab  5158  0neqopab  5816