Theorem nfop 3820

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 3627	. 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 2347	. . . 4 |- F/ x A e. _V
4		nfop.2	. . . . 5 |- F/_ x B
5	4	nfel1 2347	. . . 4 |- F/ x B e. _V
6	2	nfsn 3678	. . . . . 6 |- F/_ x { A }
7	2, 4	nfpr 3668	. . . . . 6 |- F/_ x { A , B }
8	6, 7	nfpr 3668	. . . . 5 |- F/_ x { { A } , { A , B } }
9	8	nfcri 2330	. . . 4 |- F/ x y e. { { A } , { A , B } }
10	3, 5, 9	nf3an 1577	. . 3 |- F/ x ( A e. _V /\ B e. _V /\ y e. { { A } , { A , B } } )
11	10	nfab 2341	. 2 |- F/_ x { y | ( A e. _V /\ B e. _V /\ y e. { { A } , { A , B } } ) }
12	1, 11	nfcxfr 2333	1 |- F/_ x <. A , B >.

Syntax hints:   /\ w3a 980   e. wcel 2164   {cab 2179   F/_wnfc 2323   _Vcvv 2760   {csn 3618   {cpr 3619   <.cop 3621

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 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2175

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-v 2762  df-un 3157  df-sn 3624  df-pr 3625  df-op 3627

This theorem is referenced by:  nfopd  3821  moop2  4280  fliftfuns  5841  dfmpo  6276  qliftfuns  6673  xpf1o  6900  caucvgprprlemaddq  7768  nfseq  10528  txcnp  14439  cnmpt1t  14453  cnmpt2t  14461