Theorem nfop 3904

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 3703	. 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 2397	. . . 4 |- F/ x A e. _V
4		nfop.2	. . . . 5 |- F/_ x B
5	4	nfel1 2397	. . . 4 |- F/ x B e. _V
6	2	nfsn 3754	. . . . . 6 |- F/_ x { A }
7	2, 4	nfpr 3744	. . . . . 6 |- F/_ x { A , B }
8	6, 7	nfpr 3744	. . . . 5 |- F/_ x { { A } , { A , B } }
9	8	nfcri 2380	. . . 4 |- F/ x y e. { { A } , { A , B } }
10	3, 5, 9	nf3an 1615	. . 3 |- F/ x ( A e. _V /\ B e. _V /\ y e. { { A } , { A , B } } )
11	10	nfab 2391	. 2 |- F/_ x { y | ( A e. _V /\ B e. _V /\ y e. { { A } , { A , B } } ) }
12	1, 11	nfcxfr 2383	1 |- F/_ x <. A , B >.

Syntax hints:   /\ w3a 1005   e. wcel 2205   {cab 2220   F/_wnfc 2373   _Vcvv 2815   {csn 3694   {cpr 3695   <.cop 3697

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-ext 2216

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1812  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-v 2817  df-un 3218  df-sn 3700  df-pr 3701  df-op 3703

This theorem is referenced by:  nfopd  3905  moop2  4373  fliftfuns  5977  dfmpo  6432  qliftfuns  6866  xpf1o  7110  caucvgprprlemaddq  8039  nfseq  10843  txcnp  15262  cnmpt1t  15276  cnmpt2t  15284