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Theorem nfpr 4044
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 4011 . 2  |-  { A ,  B }  =  {
y  |  ( y  =  A  \/  y  =  B ) }
2 nfpr.1 . . . . 5  |-  F/_ x A
32nfeq2 2601 . . . 4  |-  F/ x  y  =  A
4 nfpr.2 . . . . 5  |-  F/_ x B
54nfeq2 2601 . . . 4  |-  F/ x  y  =  B
63, 5nfor 1991 . . 3  |-  F/ x
( y  =  A  \/  y  =  B )
76nfab 2588 . 2  |-  F/_ x { y  |  ( y  =  A  \/  y  =  B ) }
81, 7nfcxfr 2582 1  |-  F/_ x { A ,  B }
Colors of variables: wff setvar class
Syntax hints:    \/ wo 369    = wceq 1437   {cab 2407   F/_wnfc 2570   {cpr 3998
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1748  ax-6 1794  ax-7 1839  ax-10 1887  ax-11 1892  ax-12 1905  ax-13 2053  ax-ext 2400
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1787  df-clab 2408  df-cleq 2414  df-clel 2417  df-nfc 2572  df-v 3083  df-un 3441  df-sn 3997  df-pr 3999
This theorem is referenced by:  nfsn  4054  nfop  4200  nfaltop  30739
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