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Theorem nfint 4458
 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 4449 . 2 𝐴 = {𝑦 ∣ ∀𝑧𝐴 𝑦𝑧}
2 nfint.1 . . . 4 𝑥𝐴
3 nfv 1840 . . . 4 𝑥 𝑦𝑧
42, 3nfral 2941 . . 3 𝑥𝑧𝐴 𝑦𝑧
54nfab 2765 . 2 𝑥{𝑦 ∣ ∀𝑧𝐴 𝑦𝑧}
61, 5nfcxfr 2759 1 𝑥 𝐴
 Colors of variables: wff setvar class Syntax hints:  {cab 2607  Ⅎwnfc 2748  ∀wral 2908  ∩ cint 4447 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601 This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ral 2913  df-int 4448 This theorem is referenced by:  onminsb  6961  oawordeulem  7594  nnawordex  7677  rankidb  8623  cardmin2  8784  cardaleph  8872  cardmin  9346  ldsysgenld  30046  sltval2  31563  nobndlem5  31612  aomclem8  37150
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