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Theorem fipjust 36181
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 3629 . . 3  |-  ( u  =  x  ->  (
u  i^i  v )  =  ( x  i^i  v ) )
21eleq1d 2515 . 2  |-  ( u  =  x  ->  (
( u  i^i  v
)  e.  A  <->  ( x  i^i  v )  e.  A
) )
3 ineq2 3630 . . 3  |-  ( v  =  y  ->  (
x  i^i  v )  =  ( x  i^i  y ) )
43eleq1d 2515 . 2  |-  ( v  =  y  ->  (
( x  i^i  v
)  e.  A  <->  ( x  i^i  y )  e.  A
) )
52, 4cbvral2v 3029 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 188    e. wcel 1889   A.wral 2739    i^i cin 3405
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1671  ax-4 1684  ax-5 1760  ax-6 1807  ax-7 1853  ax-10 1917  ax-11 1922  ax-12 1935  ax-13 2093  ax-ext 2433
This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-tru 1449  df-ex 1666  df-nf 1670  df-sb 1800  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2583  df-ral 2744  df-v 3049  df-in 3413
This theorem is referenced by: (None)
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