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Theorem nfint 4686
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 4678 . 2 𝐴 = {𝑦 ∣ ∀𝑧𝐴 𝑦𝑧}
2 nfint.1 . . . 4 𝑥𝐴
3 nfv 2005 . . . 4 𝑥 𝑦𝑧
42, 3nfral 3140 . . 3 𝑥𝑧𝐴 𝑦𝑧
54nfab 2960 . 2 𝑥{𝑦 ∣ ∀𝑧𝐴 𝑦𝑧}
61, 5nfcxfr 2953 1 𝑥 𝐴
Colors of variables: wff setvar class
Syntax hints:  {cab 2799  wnfc 2942  wral 3103   cint 4676
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1877  ax-4 1894  ax-5 2001  ax-6 2069  ax-7 2105  ax-9 2166  ax-10 2186  ax-11 2202  ax-12 2215  ax-13 2422  ax-ext 2791
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 866  df-tru 1641  df-ex 1860  df-nf 1864  df-sb 2062  df-clab 2800  df-cleq 2806  df-clel 2809  df-nfc 2944  df-ral 3108  df-int 4677
This theorem is referenced by:  onminsb  7232  oawordeulem  7874  nnawordex  7957  rankidb  8913  cardmin2  9110  cardaleph  9198  cardmin  9674  ldsysgenld  30554  sltval2  32135  aomclem8  38133
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