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Theorem nfop 3606
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.
StepHypRef Expression
1 df-op 3425 . 2  |-  <. A ,  B >.  =  { y  |  ( A  e. 
_V  /\  B  e.  _V  /\  y  e.  { { A } ,  { A ,  B } } ) }
2 nfop.1 . . . . 5  |-  F/_ x A
32nfel1 2233 . . . 4  |-  F/ x  A  e.  _V
4 nfop.2 . . . . 5  |-  F/_ x B
54nfel1 2233 . . . 4  |-  F/ x  B  e.  _V
62nfsn 3470 . . . . . 6  |-  F/_ x { A }
72, 4nfpr 3460 . . . . . 6  |-  F/_ x { A ,  B }
86, 7nfpr 3460 . . . . 5  |-  F/_ x { { A } ,  { A ,  B } }
98nfcri 2217 . . . 4  |-  F/ x  y  e.  { { A } ,  { A ,  B } }
103, 5, 9nf3an 1499 . . 3  |-  F/ x
( A  e.  _V  /\  B  e.  _V  /\  y  e.  { { A } ,  { A ,  B } } )
1110nfab 2227 . 2  |-  F/_ x { y  |  ( A  e.  _V  /\  B  e.  _V  /\  y  e.  { { A } ,  { A ,  B } } ) }
121, 11nfcxfr 2220 1  |-  F/_ x <. A ,  B >.
Colors of variables: wff set class
Syntax hints:    /\ w3a 920    e. wcel 1434   {cab 2069   F/_wnfc 2210   _Vcvv 2610   {csn 3416   {cpr 3417   <.cop 3419
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2065
This theorem depends on definitions:  df-bi 115  df-3an 922  df-tru 1288  df-nf 1391  df-sb 1688  df-clab 2070  df-cleq 2076  df-clel 2079  df-nfc 2212  df-v 2612  df-un 2986  df-sn 3422  df-pr 3423  df-op 3425
This theorem is referenced by:  nfopd  3607  moop2  4034  fliftfuns  5490  dfmpt2  5896  qliftfuns  6278  xpf1o  6407  caucvgprprlemaddq  7030  nfiseq  9598
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