Theorem nfopab1 3826

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 3819	. 2 |- { <. x , y >. | ph } = { z | E. x E. y ( z = <. x , y >. /\ ph ) }
2		nfe1 1385	. . 3 |- F/ x E. x E. y ( z = <. x , y >. /\ ph )
3	2	nfab 2182	. 2 |- F/_ x { z | E. x E. y ( z = <. x , y >. /\ ph ) }
4	1, 3	nfcxfr 2175	1 |- F/_ x { <. x , y >. | ph }

Syntax hints:   /\ wa 97   = wceq 1243   E.wex 1381   {cab 2026   F/_wnfc 2165   <.cop 3378   {copab 3817

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022

This theorem depends on definitions:  df-bi 110  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-opab 3819

This theorem is referenced by:  nfmpt1  3850  opelopabsb  3997  ssopab2b  4013  dmopab  4546  rnopab  4581  funopab  4935  0neqopab  5550