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Theorem nfint 3654
 Description: Bound-variable hypothesis builder for intersection. (Contributed by NM, 2-Feb-1997.) (Proof shortened by Andrew Salmon, 12-Aug-2011.)
Hypothesis
Ref Expression
nfint.1 𝑥𝐴
Assertion
Ref Expression
nfint 𝑥 𝐴

Proof of Theorem nfint
Dummy variables 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 dfint2 3646 . 2 𝐴 = {𝑦 ∣ ∀𝑧𝐴 𝑦𝑧}
2 nfint.1 . . . 4 𝑥𝐴
3 nfv 1462 . . . 4 𝑥 𝑦𝑧
42, 3nfralxy 2403 . . 3 𝑥𝑧𝐴 𝑦𝑧
54nfab 2224 . 2 𝑥{𝑦 ∣ ∀𝑧𝐴 𝑦𝑧}
61, 5nfcxfr 2217 1 𝑥 𝐴
 Colors of variables: wff set class Syntax hints:  {cab 2068  Ⅎwnfc 2207  ∀wral 2349  ∩ cint 3644 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2064 This theorem depends on definitions:  df-bi 115  df-tru 1288  df-nf 1391  df-sb 1687  df-clab 2069  df-cleq 2075  df-clel 2078  df-nfc 2209  df-ral 2354  df-int 3645 This theorem is referenced by: (None)
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