Theorem nfop 3849

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 3652	. 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 2361	. . . 4 |- F/ x A e. _V
4		nfop.2	. . . . 5 |- F/_ x B
5	4	nfel1 2361	. . . 4 |- F/ x B e. _V
6	2	nfsn 3703	. . . . . 6 |- F/_ x { A }
7	2, 4	nfpr 3693	. . . . . 6 |- F/_ x { A , B }
8	6, 7	nfpr 3693	. . . . 5 |- F/_ x { { A } , { A , B } }
9	8	nfcri 2344	. . . 4 |- F/ x y e. { { A } , { A , B } }
10	3, 5, 9	nf3an 1590	. . 3 |- F/ x ( A e. _V /\ B e. _V /\ y e. { { A } , { A , B } } )
11	10	nfab 2355	. 2 |- F/_ x { y | ( A e. _V /\ B e. _V /\ y e. { { A } , { A , B } } ) }
12	1, 11	nfcxfr 2347	1 |- F/_ x <. A , B >.

Syntax hints:   /\ w3a 981   e. wcel 2178   {cab 2193   F/_wnfc 2337   _Vcvv 2776   {csn 3643   {cpr 3644   <.cop 3646

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 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-ext 2189

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-nf 1485  df-sb 1787  df-clab 2194  df-cleq 2200  df-clel 2203  df-nfc 2339  df-v 2778  df-un 3178  df-sn 3649  df-pr 3650  df-op 3652

This theorem is referenced by:  nfopd  3850  moop2  4314  fliftfuns  5890  dfmpo  6332  qliftfuns  6729  xpf1o  6966  caucvgprprlemaddq  7856  nfseq  10639  txcnp  14858  cnmpt1t  14872  cnmpt2t  14880