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Theorem nfpr 3857
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 3832 . 2  |-  { A ,  B }  =  {
y  |  ( y  =  A  \/  y  =  B ) }
2 nfpr.1 . . . . 5  |-  F/_ x A
32nfeq2 2585 . . . 4  |-  F/ x  y  =  A
4 nfpr.2 . . . . 5  |-  F/_ x B
54nfeq2 2585 . . . 4  |-  F/ x  y  =  B
63, 5nfor 1859 . . 3  |-  F/ x
( y  =  A  \/  y  =  B )
76nfab 2578 . 2  |-  F/_ x { y  |  ( y  =  A  \/  y  =  B ) }
81, 7nfcxfr 2571 1  |-  F/_ x { A ,  B }
Colors of variables: wff set class
Syntax hints:    \/ wo 359    = wceq 1653   {cab 2424   F/_wnfc 2561   {cpr 3817
This theorem is referenced by:  nfsn  3868  nfop  4002  nfaltop  25830
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-v 2960  df-un 3327  df-sn 3822  df-pr 3823
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