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Theorem nfint 4848
 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 4840 . 2 𝐴 = {𝑦 ∣ ∀𝑧𝐴 𝑦𝑧}
2 nfint.1 . . . 4 𝑥𝐴
3 nfv 1915 . . . 4 𝑥 𝑦𝑧
42, 3nfralw 3153 . . 3 𝑥𝑧𝐴 𝑦𝑧
54nfab 2925 . 2 𝑥{𝑦 ∣ ∀𝑧𝐴 𝑦𝑧}
61, 5nfcxfr 2917 1 𝑥 𝐴
 Colors of variables: wff setvar class Syntax hints:  {cab 2735  Ⅎwnfc 2899  ∀wral 3070  ∩ cint 4838 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 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2729 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-clab 2736  df-cleq 2750  df-clel 2830  df-nfc 2901  df-ral 3075  df-int 4839 This theorem is referenced by:  onminsb  7513  oawordeulem  8190  nnawordex  8273  rankidb  9262  cardmin2  9461  cardaleph  9549  cardmin  10024  ldsysgenld  31647  sltval2  33444  aomclem8  40400
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