Theorem nfop 3586

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 3407	. 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 2229	. . . 4 |- F/ x A e. _V
4		nfop.2	. . . . 5 |- F/_ x B
5	4	nfel1 2229	. . . 4 |- F/ x B e. _V
6	2	nfsn 3452	. . . . . 6 |- F/_ x { A }
7	2, 4	nfpr 3442	. . . . . 6 |- F/_ x { A , B }
8	6, 7	nfpr 3442	. . . . 5 |- F/_ x { { A } , { A , B } }
9	8	nfcri 2213	. . . 4 |- F/ x y e. { { A } , { A , B } }
10	3, 5, 9	nf3an 1498	. . 3 |- F/ x ( A e. _V /\ B e. _V /\ y e. { { A } , { A , B } } )
11	10	nfab 2223	. 2 |- F/_ x { y | ( A e. _V /\ B e. _V /\ y e. { { A } , { A , B } } ) }
12	1, 11	nfcxfr 2216	1 |- F/_ x <. A , B >.

Syntax hints:   /\ w3a 919   e. wcel 1433   {cab 2067   F/_wnfc 2206   _Vcvv 2601   {csn 3398   {cpr 3399   <.cop 3401

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-bndl 1439  ax-4 1440  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063

This theorem depends on definitions:  df-bi 115  df-3an 921  df-tru 1287  df-nf 1390  df-sb 1686  df-clab 2068  df-cleq 2074  df-clel 2077  df-nfc 2208  df-v 2603  df-un 2977  df-sn 3404  df-pr 3405  df-op 3407

This theorem is referenced by:  nfopd  3587  moop2  4006  fliftfuns  5458  dfmpt2  5864  qliftfuns  6213  caucvgprprlemaddq  6898  nfiseq  9438