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Theorem nfop 3828
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
StepHypRef Expression
1 dfopif 3809 . 2  |-  <. A ,  B >.  =  if ( ( A  e.  _V  /\  B  e.  _V ) ,  { { A } ,  { A ,  B } } ,  (/) )
2 nfop.1 . . . . 5  |-  F/_ x A
32nfel1 2442 . . . 4  |-  F/ x  A  e.  _V
4 nfop.2 . . . . 5  |-  F/_ x B
54nfel1 2442 . . . 4  |-  F/ x  B  e.  _V
63, 5nfan 1783 . . 3  |-  F/ x
( A  e.  _V  /\  B  e.  _V )
72nfsn 3704 . . . 4  |-  F/_ x { A }
82, 4nfpr 3693 . . . 4  |-  F/_ x { A ,  B }
97, 8nfpr 3693 . . 3  |-  F/_ x { { A } ,  { A ,  B } }
10 nfcv 2432 . . 3  |-  F/_ x (/)
116, 9, 10nfif 3602 . 2  |-  F/_ x if ( ( A  e. 
_V  /\  B  e.  _V ) ,  { { A } ,  { A ,  B } } ,  (/) )
121, 11nfcxfr 2429 1  |-  F/_ x <. A ,  B >.
Colors of variables: wff set class
Syntax hints:    /\ wa 358    e. wcel 1696   F/_wnfc 2419   _Vcvv 2801   (/)c0 3468   ifcif 3578   {csn 3653   {cpr 3654   <.cop 3656
This theorem is referenced by:  nfopd  3829  moop2  4277  fliftfuns  5829  dfmpt2  6225  qliftfuns  6761  xpf1o  7039  nfseq  11072  txcnp  17330  cnmpt1t  17375  cnmpt2t  17383  flfcnp2  17718  sbcopg  24037  nfaov  28147  bnj958  29288  bnj1000  29289  bnj1446  29391  bnj1447  29392  bnj1448  29393  bnj1466  29399  bnj1467  29400  bnj1519  29411  bnj1520  29412  bnj1529  29416
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-v 2803  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-sn 3659  df-pr 3660  df-op 3662
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