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Theorem nfpr 3466
Description: Bound-variable hypothesis builder for unordered pairs. (Contributed by NM, 14-Nov-1995.)
Hypotheses
Ref Expression
nfpr.1  |-  F/_ x A
nfpr.2  |-  F/_ x B
Assertion
Ref Expression
nfpr  |-  F/_ x { A ,  B }

Proof of Theorem nfpr
Dummy variable  y is distinct from all other variables.
StepHypRef Expression
1 dfpr2 3441 . 2  |-  { A ,  B }  =  {
y  |  ( y  =  A  \/  y  =  B ) }
2 nfpr.1 . . . . 5  |-  F/_ x A
32nfeq2 2234 . . . 4  |-  F/ x  y  =  A
4 nfpr.2 . . . . 5  |-  F/_ x B
54nfeq2 2234 . . . 4  |-  F/ x  y  =  B
63, 5nfor 1507 . . 3  |-  F/ x
( y  =  A  \/  y  =  B )
76nfab 2227 . 2  |-  F/_ x { y  |  ( y  =  A  \/  y  =  B ) }
81, 7nfcxfr 2220 1  |-  F/_ x { A ,  B }
Colors of variables: wff set class
Syntax hints:    \/ wo 662    = wceq 1285   {cab 2069   F/_wnfc 2210   {cpr 3423
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-tru 1288  df-nf 1391  df-sb 1688  df-clab 2070  df-cleq 2076  df-clel 2079  df-nfc 2212  df-v 2614  df-un 2988  df-sn 3428  df-pr 3429
This theorem is referenced by:  nfsn  3476  nfop  3612
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