Theorem fipjust 37870

Description: A definition of the finite intersection property of a class based on closure under pairwise 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 3807	. . 3 |- ( u = x -> ( u i^i v ) = ( x i^i v ) )
2	1	eleq1d 2686	. 2 |- ( u = x -> ( ( u i^i v ) e. A <-> ( x i^i v ) e. A ) )
3		ineq2 3808	. . 3 |- ( v = y -> ( x i^i v ) = ( x i^i y ) )
4	3	eleq1d 2686	. 2 |- ( v = y -> ( ( x i^i v ) e. A <-> ( x i^i y ) e. A ) )
5	2, 4	cbvral2v 3179	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 196   e. wcel 1990   A.wral 2912   i^i cin 3573

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-v 3202  df-in 3581

This theorem is referenced by: (None)