Theorem nfop 3565

Description: Bound-variable hypothesis builder for ordered pairs. (Contributed by NM, 14-Nov-1995.)

Hypotheses

Ref	Expression
nfop.1	|- F/_ x A
nfop.2	|- F/_ x B

Assertion

Ref	Expression
nfop	|- F/_ x <. A , B >.

Proof of Theorem nfop

Dummy variable y is distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-op 3384	. 2 |- <. A , B >. = { y | ( A e. _V /\ B e. _V /\ y e. { { A } , { A , B } } ) }
2		nfop.1	. . . . 5 |- F/_ x A
3	2	nfel1 2188	. . . 4 |- F/ x A e. _V
4		nfop.2	. . . . 5 |- F/_ x B
5	4	nfel1 2188	. . . 4 |- F/ x B e. _V
6	2	nfsn 3430	. . . . . 6 |- F/_ x { A }
7	2, 4	nfpr 3420	. . . . . 6 |- F/_ x { A , B }
8	6, 7	nfpr 3420	. . . . 5 |- F/_ x { { A } , { A , B } }
9	8	nfcri 2172	. . . 4 |- F/ x y e. { { A } , { A , B } }
10	3, 5, 9	nf3an 1458	. . 3 |- F/ x ( A e. _V /\ B e. _V /\ y e. { { A } , { A , B } } )
11	10	nfab 2182	. 2 |- F/_ x { y | ( A e. _V /\ B e. _V /\ y e. { { A } , { A , B } } ) }
12	1, 11	nfcxfr 2175	1 |- F/_ x <. A , B >.

Syntax hints:   /\ w3a 885   e. wcel 1393   {cab 2026   F/_wnfc 2165   _Vcvv 2557   {csn 3375   {cpr 3376   <.cop 3378

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-3an 887  df-tru 1246  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-v 2559  df-un 2922  df-sn 3381  df-pr 3382  df-op 3384

This theorem is referenced by:  nfopd  3566  moop2  3988  fliftfuns  5438  dfmpt2  5844  qliftfuns  6190  caucvgprprlemaddq  6806  nfiseq  9218