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

Step	Hyp	Ref	Expression
1		ineq1 3689	. . 3 |- ( u = x -> ( u i^i v ) = ( x i^i v ) )
2	1	eleq1d 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 ) )
4	3	eleq1d 2526	. 2 |- ( v = y -> ( ( x i^i v ) e. A <-> ( x i^i y ) e. A ) )
5	2, 4	cbvral2v 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 )

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)