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Theorem nfpr 3411
 Description: Bound-variable hypothesis builder for unordered pairs. (Contributed by NM, 14-Nov-1995.)
Hypotheses
Ref Expression
nfpr.1 xA
nfpr.2 xB
Assertion
Ref Expression
nfpr x{A, B}

Proof of Theorem nfpr
Dummy variable y is distinct from all other variables.
StepHypRef Expression
1 dfpr2 3383 . 2 {A, B} = {y ∣ (y = A y = B)}
2 nfpr.1 . . . . 5 xA
32nfeq2 2186 . . . 4 x y = A
4 nfpr.2 . . . . 5 xB
54nfeq2 2186 . . . 4 x y = B
63, 5nfor 1463 . . 3 x(y = A y = B)
76nfab 2179 . 2 x{y ∣ (y = A y = B)}
81, 7nfcxfr 2172 1 x{A, B}
 Colors of variables: wff set class Syntax hints:   ∨ wo 628   = wceq 1242  {cab 2023  Ⅎwnfc 2162  {cpr 3368 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bnd 1396  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019 This theorem depends on definitions:  df-bi 110  df-tru 1245  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-v 2553  df-un 2916  df-sn 3373  df-pr 3374 This theorem is referenced by:  nfsn  3421  nfop  3556
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