Theorem nfop 3873

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 3675	. 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 2383	. . . 4 |- F/ x A e. _V
4		nfop.2	. . . . 5 |- F/_ x B
5	4	nfel1 2383	. . . 4 |- F/ x B e. _V
6	2	nfsn 3726	. . . . . 6 |- F/_ x { A }
7	2, 4	nfpr 3716	. . . . . 6 |- F/_ x { A , B }
8	6, 7	nfpr 3716	. . . . 5 |- F/_ x { { A } , { A , B } }
9	8	nfcri 2366	. . . 4 |- F/ x y e. { { A } , { A , B } }
10	3, 5, 9	nf3an 1612	. . 3 |- F/ x ( A e. _V /\ B e. _V /\ y e. { { A } , { A , B } } )
11	10	nfab 2377	. 2 |- F/_ x { y | ( A e. _V /\ B e. _V /\ y e. { { A } , { A , B } } ) }
12	1, 11	nfcxfr 2369	1 |- F/_ x <. A , B >.

Syntax hints:   /\ w3a 1002   e. wcel 2200   {cab 2215   F/_wnfc 2359   _Vcvv 2799   {csn 3666   {cpr 3667   <.cop 3669

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 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-ext 2211

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-nf 1507  df-sb 1809  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-v 2801  df-un 3201  df-sn 3672  df-pr 3673  df-op 3675

This theorem is referenced by:  nfopd  3874  moop2  4338  fliftfuns  5922  dfmpo  6369  qliftfuns  6766  xpf1o  7005  caucvgprprlemaddq  7895  nfseq  10679  txcnp  14945  cnmpt1t  14959  cnmpt2t  14967