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Theorem nfop 3758
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 3569 . 2  |-  <. A ,  B >.  =  { y  |  ( A  e. 
_V  /\  B  e.  _V  /\  y  e.  { { A } ,  { A ,  B } } ) }
2 nfop.1 . . . . 5  |-  F/_ x A
32nfel1 2310 . . . 4  |-  F/ x  A  e.  _V
4 nfop.2 . . . . 5  |-  F/_ x B
54nfel1 2310 . . . 4  |-  F/ x  B  e.  _V
62nfsn 3620 . . . . . 6  |-  F/_ x { A }
72, 4nfpr 3610 . . . . . 6  |-  F/_ x { A ,  B }
86, 7nfpr 3610 . . . . 5  |-  F/_ x { { A } ,  { A ,  B } }
98nfcri 2293 . . . 4  |-  F/ x  y  e.  { { A } ,  { A ,  B } }
103, 5, 9nf3an 1546 . . 3  |-  F/ x
( A  e.  _V  /\  B  e.  _V  /\  y  e.  { { A } ,  { A ,  B } } )
1110nfab 2304 . 2  |-  F/_ x { y  |  ( A  e.  _V  /\  B  e.  _V  /\  y  e.  { { A } ,  { A ,  B } } ) }
121, 11nfcxfr 2296 1  |-  F/_ x <. A ,  B >.
Colors of variables: wff set class
Syntax hints:    /\ w3a 963    e. wcel 2128   {cab 2143   F/_wnfc 2286   _Vcvv 2712   {csn 3560   {cpr 3561   <.cop 3563
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2139
This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1338  df-nf 1441  df-sb 1743  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-v 2714  df-un 3106  df-sn 3566  df-pr 3567  df-op 3569
This theorem is referenced by:  nfopd  3759  moop2  4212  fliftfuns  5749  dfmpo  6171  qliftfuns  6565  xpf1o  6790  caucvgprprlemaddq  7629  nfseq  10358  txcnp  12713  cnmpt1t  12727  cnmpt2t  12735
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