Theorem nfop 3915

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 3714	. 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 2403	. . . 4 |- F/ x A e. _V
4		nfop.2	. . . . 5 |- F/_ x B
5	4	nfel1 2403	. . . 4 |- F/ x B e. _V
6	2	nfsn 3765	. . . . . 6 |- F/_ x { A }
7	2, 4	nfpr 3755	. . . . . 6 |- F/_ x { A , B }
8	6, 7	nfpr 3755	. . . . 5 |- F/_ x { { A } , { A , B } }
9	8	nfcri 2386	. . . 4 |- F/ x y e. { { A } , { A , B } }
10	3, 5, 9	nf3an 1619	. . 3 |- F/ x ( A e. _V /\ B e. _V /\ y e. { { A } , { A , B } } )
11	10	nfab 2397	. 2 |- F/_ x { y | ( A e. _V /\ B e. _V /\ y e. { { A } , { A , B } } ) }
12	1, 11	nfcxfr 2389	1 |- F/_ x <. A , B >.

Syntax hints:   /\ w3a 1009   e. wcel 2209   {cab 2224   F/_wnfc 2379   _Vcvv 2821   {csn 3705   {cpr 3706   <.cop 3708

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 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-ext 2220

This theorem depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-nf 1514  df-sb 1816  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-v 2823  df-un 3224  df-sn 3711  df-pr 3712  df-op 3714

This theorem is referenced by:  nfopd  3916  moop2  4387  fliftfuns  5994  dfmpo  6449  qliftfuns  6883  xpf1o  7134  caucvgprprlemaddq  8065  nfseq  10872  txcnp  15295  cnmpt1t  15309  cnmpt2t  15317