Users' Mathboxes Mathbox for Richard Penner < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  fipjust Structured version   Unicode version

Theorem fipjust 37953
Description: A definition of the finite intersection property of a class based on closure under pair-wise intersection of its elements is independent of the dummy variables. (Contributed by Richard Penner, 1-Jan-2020.)
Assertion
Ref Expression
fipjust  |-  ( A. u  e.  A  A. v  e.  A  (
u  i^i  v )  e.  A  <->  A. x  e.  A  A. y  e.  A  ( x  i^i  y
)  e.  A )
Distinct variable group:    v, u, x, y, A

Proof of Theorem fipjust
StepHypRef Expression
1 ineq1 3689 . . 3  |-  ( u  =  x  ->  (
u  i^i  v )  =  ( x  i^i  v ) )
21eleq1d 2526 . 2  |-  ( u  =  x  ->  (
( u  i^i  v
)  e.  A  <->  ( x  i^i  v )  e.  A
) )
3 ineq2 3690 . . 3  |-  ( v  =  y  ->  (
x  i^i  v )  =  ( x  i^i  y ) )
43eleq1d 2526 . 2  |-  ( v  =  y  ->  (
( x  i^i  v
)  e.  A  <->  ( x  i^i  y )  e.  A
) )
52, 4cbvral2v 3092 1  |-  ( A. u  e.  A  A. v  e.  A  (
u  i^i  v )  e.  A  <->  A. x  e.  A  A. y  e.  A  ( x  i^i  y
)  e.  A )
Colors of variables: wff setvar class
Syntax hints:    <-> wb 184    e. wcel 1819   A.wral 2807    i^i cin 3470
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ral 2812  df-v 3111  df-in 3478
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator