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Theorem nfint 4915
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 4907 . 2 𝐴 = {𝑦 ∣ ∀𝑧𝐴 𝑦𝑧}
2 nfint.1 . . . 4 𝑥𝐴
3 nfv 1917 . . . 4 𝑥 𝑦𝑧
42, 3nfralw 3292 . . 3 𝑥𝑧𝐴 𝑦𝑧
54nfab 2911 . 2 𝑥{𝑦 ∣ ∀𝑧𝐴 𝑦𝑧}
61, 5nfcxfr 2903 1 𝑥 𝐴
Colors of variables: wff setvar class
Syntax hints:  {cab 2713  wnfc 2885  wral 3062   cint 4905
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2707
This theorem depends on definitions:  df-bi 206  df-an 397  df-ex 1782  df-nf 1786  df-sb 2068  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2887  df-ral 3063  df-int 4906
This theorem is referenced by:  onminsb  7725  oawordeulem  8497  nnawordex  8580  rankidb  9732  cardmin2  9931  cardaleph  10021  cardmin  10496  sltval2  26988  ldsysgenld  32628  aomclem8  41326
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