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Theorem nfint 3751
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 3743 . 2 𝐴 = {𝑦 ∣ ∀𝑧𝐴 𝑦𝑧}
2 nfint.1 . . . 4 𝑥𝐴
3 nfv 1493 . . . 4 𝑥 𝑦𝑧
42, 3nfralxy 2448 . . 3 𝑥𝑧𝐴 𝑦𝑧
54nfab 2263 . 2 𝑥{𝑦 ∣ ∀𝑧𝐴 𝑦𝑧}
61, 5nfcxfr 2255 1 𝑥 𝐴
Colors of variables: wff set class
Syntax hints:  {cab 2103  wnfc 2245  wral 2393   cint 3741
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 683  ax-5 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-10 1468  ax-11 1469  ax-i12 1470  ax-bndl 1471  ax-4 1472  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500  ax-ext 2099
This theorem depends on definitions:  df-bi 116  df-tru 1319  df-nf 1422  df-sb 1721  df-clab 2104  df-cleq 2110  df-clel 2113  df-nfc 2247  df-ral 2398  df-int 3742
This theorem is referenced by: (None)
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