Theorem nfopab1 4051

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 4044	. 2 |- { <. x , y >. | ph } = { z | E. x E. y ( z = <. x , y >. /\ ph ) }
2		nfe1 1484	. . 3 |- F/ x E. x E. y ( z = <. x , y >. /\ ph )
3	2	nfab 2313	. 2 |- F/_ x { z | E. x E. y ( z = <. x , y >. /\ ph ) }
4	1, 3	nfcxfr 2305	1 |- F/_ x { <. x , y >. | ph }

Syntax hints:   /\ wa 103   = wceq 1343   E.wex 1480   {cab 2151   F/_wnfc 2295   <.cop 3579   {copab 4042

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 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-ext 2147

This theorem depends on definitions:  df-bi 116  df-nf 1449  df-sb 1751  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-opab 4044

This theorem is referenced by:  nfmpt1  4075  opelopabsb  4238  ssopab2b  4254  dmopab  4815  rnopab  4851  funopab  5223  0neqopab  5887