Theorem nfop 3825

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 3632	. 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 2350	. . . 4 |- F/ x A e. _V
4		nfop.2	. . . . 5 |- F/_ x B
5	4	nfel1 2350	. . . 4 |- F/ x B e. _V
6	2	nfsn 3683	. . . . . 6 |- F/_ x { A }
7	2, 4	nfpr 3673	. . . . . 6 |- F/_ x { A , B }
8	6, 7	nfpr 3673	. . . . 5 |- F/_ x { { A } , { A , B } }
9	8	nfcri 2333	. . . 4 |- F/ x y e. { { A } , { A , B } }
10	3, 5, 9	nf3an 1580	. . 3 |- F/ x ( A e. _V /\ B e. _V /\ y e. { { A } , { A , B } } )
11	10	nfab 2344	. 2 |- F/_ x { y | ( A e. _V /\ B e. _V /\ y e. { { A } , { A , B } } ) }
12	1, 11	nfcxfr 2336	1 |- F/_ x <. A , B >.

Syntax hints:   /\ w3a 980   e. wcel 2167   {cab 2182   F/_wnfc 2326   _Vcvv 2763   {csn 3623   {cpr 3624   <.cop 3626

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 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-ext 2178

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1475  df-sb 1777  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-v 2765  df-un 3161  df-sn 3629  df-pr 3630  df-op 3632

This theorem is referenced by:  nfopd  3826  moop2  4285  fliftfuns  5848  dfmpo  6290  qliftfuns  6687  xpf1o  6914  caucvgprprlemaddq  7792  nfseq  10566  txcnp  14591  cnmpt1t  14605  cnmpt2t  14613