Theorem nfop 3835

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 3642	. 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 2359	. . . 4 |- F/ x A e. _V
4		nfop.2	. . . . 5 |- F/_ x B
5	4	nfel1 2359	. . . 4 |- F/ x B e. _V
6	2	nfsn 3693	. . . . . 6 |- F/_ x { A }
7	2, 4	nfpr 3683	. . . . . 6 |- F/_ x { A , B }
8	6, 7	nfpr 3683	. . . . 5 |- F/_ x { { A } , { A , B } }
9	8	nfcri 2342	. . . 4 |- F/ x y e. { { A } , { A , B } }
10	3, 5, 9	nf3an 1589	. . 3 |- F/ x ( A e. _V /\ B e. _V /\ y e. { { A } , { A , B } } )
11	10	nfab 2353	. 2 |- F/_ x { y | ( A e. _V /\ B e. _V /\ y e. { { A } , { A , B } } ) }
12	1, 11	nfcxfr 2345	1 |- F/_ x <. A , B >.

Syntax hints:   /\ w3a 981   e. wcel 2176   {cab 2191   F/_wnfc 2335   _Vcvv 2772   {csn 3633   {cpr 3634   <.cop 3636

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 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-10 1528  ax-11 1529  ax-i12 1530  ax-bndl 1532  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-i5r 1558  ax-ext 2187

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-nf 1484  df-sb 1786  df-clab 2192  df-cleq 2198  df-clel 2201  df-nfc 2337  df-v 2774  df-un 3170  df-sn 3639  df-pr 3640  df-op 3642

This theorem is referenced by:  nfopd  3836  moop2  4297  fliftfuns  5869  dfmpo  6311  qliftfuns  6708  xpf1o  6943  caucvgprprlemaddq  7823  nfseq  10604  txcnp  14776  cnmpt1t  14790  cnmpt2t  14798