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Theorem nfop 3781
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 3592 . 2  |-  <. A ,  B >.  =  { y  |  ( A  e. 
_V  /\  B  e.  _V  /\  y  e.  { { A } ,  { A ,  B } } ) }
2 nfop.1 . . . . 5  |-  F/_ x A
32nfel1 2323 . . . 4  |-  F/ x  A  e.  _V
4 nfop.2 . . . . 5  |-  F/_ x B
54nfel1 2323 . . . 4  |-  F/ x  B  e.  _V
62nfsn 3643 . . . . . 6  |-  F/_ x { A }
72, 4nfpr 3633 . . . . . 6  |-  F/_ x { A ,  B }
86, 7nfpr 3633 . . . . 5  |-  F/_ x { { A } ,  { A ,  B } }
98nfcri 2306 . . . 4  |-  F/ x  y  e.  { { A } ,  { A ,  B } }
103, 5, 9nf3an 1559 . . 3  |-  F/ x
( A  e.  _V  /\  B  e.  _V  /\  y  e.  { { A } ,  { A ,  B } } )
1110nfab 2317 . 2  |-  F/_ x { y  |  ( A  e.  _V  /\  B  e.  _V  /\  y  e.  { { A } ,  { A ,  B } } ) }
121, 11nfcxfr 2309 1  |-  F/_ x <. A ,  B >.
Colors of variables: wff set class
Syntax hints:    /\ w3a 973    e. wcel 2141   {cab 2156   F/_wnfc 2299   _Vcvv 2730   {csn 3583   {cpr 3584   <.cop 3586
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 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-ext 2152
This theorem depends on definitions:  df-bi 116  df-3an 975  df-tru 1351  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-v 2732  df-un 3125  df-sn 3589  df-pr 3590  df-op 3592
This theorem is referenced by:  nfopd  3782  moop2  4236  fliftfuns  5777  dfmpo  6202  qliftfuns  6597  xpf1o  6822  caucvgprprlemaddq  7670  nfseq  10411  txcnp  13065  cnmpt1t  13079  cnmpt2t  13087
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