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Theorem nfpr 3770
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 3745 . 2  |-  { A ,  B }  =  {
y  |  ( y  =  A  \/  y  =  B ) }
2 nfpr.1 . . . . 5  |-  F/_ x A
32nfeq2 2513 . . . 4  |-  F/ x  y  =  A
4 nfpr.2 . . . . 5  |-  F/_ x B
54nfeq2 2513 . . . 4  |-  F/ x  y  =  B
63, 5nfor 1846 . . 3  |-  F/ x
( y  =  A  \/  y  =  B )
76nfab 2506 . 2  |-  F/_ x { y  |  ( y  =  A  \/  y  =  B ) }
81, 7nfcxfr 2499 1  |-  F/_ x { A ,  B }
Colors of variables: wff set class
Syntax hints:    \/ wo 357    = wceq 1647   {cab 2352   F/_wnfc 2489   {cpr 3730
This theorem is referenced by:  nfsn  3781  nfop  3914  nfaltop  25256
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1551  ax-5 1562  ax-17 1621  ax-9 1659  ax-8 1680  ax-6 1734  ax-7 1739  ax-11 1751  ax-12 1937  ax-ext 2347
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1324  df-ex 1547  df-nf 1550  df-sb 1654  df-clab 2353  df-cleq 2359  df-clel 2362  df-nfc 2491  df-v 2875  df-un 3243  df-sn 3735  df-pr 3736
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