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Theorem nfpr 3823
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 3798 . 2  |-  { A ,  B }  =  {
y  |  ( y  =  A  \/  y  =  B ) }
2 nfpr.1 . . . . 5  |-  F/_ x A
32nfeq2 2559 . . . 4  |-  F/ x  y  =  A
4 nfpr.2 . . . . 5  |-  F/_ x B
54nfeq2 2559 . . . 4  |-  F/ x  y  =  B
63, 5nfor 1854 . . 3  |-  F/ x
( y  =  A  \/  y  =  B )
76nfab 2552 . 2  |-  F/_ x { y  |  ( y  =  A  \/  y  =  B ) }
81, 7nfcxfr 2545 1  |-  F/_ x { A ,  B }
Colors of variables: wff set class
Syntax hints:    \/ wo 358    = wceq 1649   {cab 2398   F/_wnfc 2535   {cpr 3783
This theorem is referenced by:  nfsn  3834  nfop  3968  nfaltop  25737
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2393
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-clab 2399  df-cleq 2405  df-clel 2408  df-nfc 2537  df-v 2926  df-un 3293  df-sn 3788  df-pr 3789
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