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	⊢ ( ∀ 𝑢 ∈ 𝐴 ∀ 𝑣 ∈ 𝐴 ( 𝑢 ∩ 𝑣 ) ∈ 𝐴 ↔ ∀ 𝑥 ∈ 𝐴 ∀ 𝑦 ∈ 𝐴 ( 𝑥 ∩ 𝑦 ) ∈ 𝐴 )

Proof

Step	Hyp	Ref	Expression
1		ineq1	⊢ ( 𝑢 = 𝑥 → ( 𝑢 ∩ 𝑣 ) = ( 𝑥 ∩ 𝑣 ) )
2	1	eleq1d	⊢ ( 𝑢 = 𝑥 → ( ( 𝑢 ∩ 𝑣 ) ∈ 𝐴 ↔ ( 𝑥 ∩ 𝑣 ) ∈ 𝐴 ) )
3		ineq2	⊢ ( 𝑣 = 𝑦 → ( 𝑥 ∩ 𝑣 ) = ( 𝑥 ∩ 𝑦 ) )
4	3	eleq1d	⊢ ( 𝑣 = 𝑦 → ( ( 𝑥 ∩ 𝑣 ) ∈ 𝐴 ↔ ( 𝑥 ∩ 𝑦 ) ∈ 𝐴 ) )
5	2 4	cbvral2vw	⊢ ( ∀ 𝑢 ∈ 𝐴 ∀ 𝑣 ∈ 𝐴 ( 𝑢 ∩ 𝑣 ) ∈ 𝐴 ↔ ∀ 𝑥 ∈ 𝐴 ∀ 𝑦 ∈ 𝐴 ( 𝑥 ∩ 𝑦 ) ∈ 𝐴 )