Theorem fipjust

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 RP, 1-Jan-2020)

	Ref	Expression
Assertion	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 )

Proof

Step	Hyp	Ref	Expression
1		ineq1	|- ( u = x -> ( u i^i v ) = ( x i^i v ) )
2	1	eleq1d	|- ( u = x -> ( ( u i^i v ) e. A <-> ( x i^i v ) e. A ) )
3		ineq2	|- ( v = y -> ( x i^i v ) = ( x i^i y ) )
4	3	eleq1d	|- ( v = y -> ( ( x i^i v ) e. A <-> ( x i^i y ) e. A ) )
5	2 4	cbvral2vw	|- ( 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 )