Metamath Proof Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >  nfint Structured version   Visualization version   GIF version

Theorem nfint 4626
 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 4617 . 2 𝐴 = {𝑦 ∣ ∀𝑧𝐴 𝑦𝑧}
2 nfint.1 . . . 4 𝑥𝐴
3 nfv 1980 . . . 4 𝑥 𝑦𝑧
42, 3nfral 3071 . . 3 𝑥𝑧𝐴 𝑦𝑧
54nfab 2895 . 2 𝑥{𝑦 ∣ ∀𝑧𝐴 𝑦𝑧}
61, 5nfcxfr 2888 1 𝑥 𝐴
 Colors of variables: wff setvar class Syntax hints:  {cab 2734  Ⅎwnfc 2877  ∀wral 3038  ∩ cint 4615 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1859  ax-4 1874  ax-5 1976  ax-6 2042  ax-7 2078  ax-9 2136  ax-10 2156  ax-11 2171  ax-12 2184  ax-13 2379  ax-ext 2728 This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-tru 1623  df-ex 1842  df-nf 1847  df-sb 2035  df-clab 2735  df-cleq 2741  df-clel 2744  df-nfc 2879  df-ral 3043  df-int 4616 This theorem is referenced by:  onminsb  7152  oawordeulem  7791  nnawordex  7874  rankidb  8824  cardmin2  8985  cardaleph  9073  cardmin  9549  ldsysgenld  30503  sltval2  32086  aomclem8  38102
 Copyright terms: Public domain W3C validator