Theorem nfopab1 4102

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 4095	. 2 |- { <. x , y >. | ph } = { z | E. x E. y ( z = <. x , y >. /\ ph ) }
2		nfe1 1510	. . 3 |- F/ x E. x E. y ( z = <. x , y >. /\ ph )
3	2	nfab 2344	. 2 |- F/_ x { z | E. x E. y ( z = <. x , y >. /\ ph ) }
4	1, 3	nfcxfr 2336	1 |- F/_ x { <. x , y >. | ph }

Syntax hints:   /\ wa 104   = wceq 1364   E.wex 1506   {cab 2182   F/_wnfc 2326   <.cop 3625   {copab 4093

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 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-ext 2178

This theorem depends on definitions:  df-bi 117  df-nf 1475  df-sb 1777  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-opab 4095

This theorem is referenced by:  nfmpt1  4126  opelopabsb  4294  ssopab2b  4311  dmopab  4877  rnopab  4913  funopab  5293  0neqopab  5967