Theorem nfopab2 4115

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 4107	. 2 |- { <. x , y >. | ph } = { z | E. x E. y ( z = <. x , y >. /\ ph ) }
2		nfe1 1519	. . . 4 |- F/ y E. y ( z = <. x , y >. /\ ph )
3	2	nfex 1660	. . 3 |- F/ y E. x E. y ( z = <. x , y >. /\ ph )
4	3	nfab 2353	. 2 |- F/_ y { z | E. x E. y ( z = <. x , y >. /\ ph ) }
5	1, 4	nfcxfr 2345	1 |- F/_ y { <. x , y >. | ph }

Syntax hints:   /\ wa 104   = wceq 1373   E.wex 1515   {cab 2191   F/_wnfc 2335   <.cop 3636   {copab 4105

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 711  ax-5 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-10 1528  ax-11 1529  ax-i12 1530  ax-bndl 1532  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-i5r 1558  ax-ext 2187

This theorem depends on definitions:  df-bi 117  df-nf 1484  df-sb 1786  df-clab 2192  df-cleq 2198  df-clel 2201  df-nfc 2337  df-opab 4107

This theorem is referenced by:  opelopabsb  4307  ssopab2b  4324  dmopab  4890  rnopab  4926  funopab  5307  0neqopab  5992