Theorem fipjust 43520

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.)

Assertion

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

Distinct variable group:   𝑣,𝑢,𝑥,𝑦,𝐴

Proof of Theorem fipjust

Step	Hyp	Ref	Expression
1		ineq1 4221	. . 3 ⊢ (𝑢 = 𝑥 → (𝑢 ∩ 𝑣) = (𝑥 ∩ 𝑣))
2	1	eleq1d 2822	. 2 ⊢ (𝑢 = 𝑥 → ((𝑢 ∩ 𝑣) ∈ 𝐴 ↔ (𝑥 ∩ 𝑣) ∈ 𝐴))
3		ineq2 4222	. . 3 ⊢ (𝑣 = 𝑦 → (𝑥 ∩ 𝑣) = (𝑥 ∩ 𝑦))
4	3	eleq1d 2822	. 2 ⊢ (𝑣 = 𝑦 → ((𝑥 ∩ 𝑣) ∈ 𝐴 ↔ (𝑥 ∩ 𝑦) ∈ 𝐴))
5	2, 4	cbvral2vw 3237	1 ⊢ (∀𝑢 ∈ 𝐴 ∀𝑣 ∈ 𝐴 (𝑢 ∩ 𝑣) ∈ 𝐴 ↔ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥 ∩ 𝑦) ∈ 𝐴)

Syntax hints:   ↔ wb 206   ∈ wcel 2104  ∀wral 3057   ∩ cin 3962

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1963  ax-7 2003  ax-8 2106  ax-9 2114  ax-ext 2704

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1538  df-ex 1775  df-sb 2061  df-clab 2711  df-cleq 2725  df-clel 2812  df-ral 3058  df-rab 3433  df-in 3970

This theorem is referenced by: (None)