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Theorem nfop 4024
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 4005 . 2  |-  <. A ,  B >.  =  if ( ( A  e.  _V  /\  B  e.  _V ) ,  { { A } ,  { A ,  B } } ,  (/) )
2 nfop.1 . . . . 5  |-  F/_ x A
32nfel1 2588 . . . 4  |-  F/ x  A  e.  _V
4 nfop.2 . . . . 5  |-  F/_ x B
54nfel1 2588 . . . 4  |-  F/ x  B  e.  _V
63, 5nfan 1848 . . 3  |-  F/ x
( A  e.  _V  /\  B  e.  _V )
72nfsn 3890 . . . 4  |-  F/_ x { A }
82, 4nfpr 3879 . . . 4  |-  F/_ x { A ,  B }
97, 8nfpr 3879 . . 3  |-  F/_ x { { A } ,  { A ,  B } }
10 nfcv 2578 . . 3  |-  F/_ x (/)
116, 9, 10nfif 3787 . 2  |-  F/_ x if ( ( A  e. 
_V  /\  B  e.  _V ) ,  { { A } ,  { A ,  B } } ,  (/) )
121, 11nfcxfr 2575 1  |-  F/_ x <. A ,  B >.
Colors of variables: wff set class
Syntax hints:    /\ wa 360    e. wcel 1727   F/_wnfc 2565   _Vcvv 2962   (/)c0 3613   ifcif 3763   {csn 3838   {cpr 3839   <.cop 3841
This theorem is referenced by:  nfopd  4025  moop2  4480  fliftfuns  6065  dfmpt2  6466  qliftfuns  7020  xpf1o  7298  nfseq  11364  txcnp  17683  cnmpt1t  17728  cnmpt2t  17736  flfcnp2  18070  sbcopg  25157  nfaov  28057  bnj958  29409  bnj1000  29410  bnj1446  29512  bnj1447  29513  bnj1448  29514  bnj1466  29520  bnj1467  29521  bnj1519  29532  bnj1520  29533  bnj1529  29537
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1668  ax-8 1689  ax-6 1746  ax-7 1751  ax-11 1763  ax-12 1953  ax-ext 2423
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2567  df-v 2964  df-dif 3309  df-un 3311  df-in 3313  df-ss 3320  df-nul 3614  df-if 3764  df-sn 3844  df-pr 3845  df-op 3847
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