Theorem nfopab2 4164

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 4156	. 2 |- { <. x , y >. | ph } = { z | E. x E. y ( z = <. x , y >. /\ ph ) }
2		nfe1 1545	. . . 4 |- F/ y E. y ( z = <. x , y >. /\ ph )
3	2	nfex 1686	. . 3 |- F/ y E. x E. y ( z = <. x , y >. /\ ph )
4	3	nfab 2380	. 2 |- F/_ y { z | E. x E. y ( z = <. x , y >. /\ ph ) }
5	1, 4	nfcxfr 2372	1 |- F/_ y { <. x , y >. | ph }

Syntax hints:   /\ wa 104   = wceq 1398   E.wex 1541   {cab 2217   F/_wnfc 2362   <.cop 3676   {copab 4154

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-ext 2213

This theorem depends on definitions:  df-bi 117  df-nf 1510  df-sb 1811  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-opab 4156

This theorem is referenced by:  opelopabsb  4360  ssopab2b  4377  dmopab  4948  rnopab  4985  funopab  5368  0neqopab  6076