Theorem iniin2 44364

Description: Indexed intersection of intersection. (Contributed by Glauco Siliprandi, 23-Oct-2021.)

Assertion

Ref	Expression
iniin2	⊢ (𝐴 ≠ ∅ → (𝐵 ∩ ∩ 𝑥 ∈ 𝐴 𝐶) = ∩ 𝑥 ∈ 𝐴 (𝐵 ∩ 𝐶))

Distinct variable groups:   𝑥,𝐴   𝑥,𝐵

Allowed substitution hint:   𝐶(𝑥)

Proof of Theorem iniin2

Step	Hyp	Ref	Expression
1		iinin2 5072	. 2 ⊢ (𝐴 ≠ ∅ → ∩ 𝑥 ∈ 𝐴 (𝐵 ∩ 𝐶) = (𝐵 ∩ ∩ 𝑥 ∈ 𝐴 𝐶))
2	1	eqcomd 2730	1 ⊢ (𝐴 ≠ ∅ → (𝐵 ∩ ∩ 𝑥 ∈ 𝐴 𝐶) = ∩ 𝑥 ∈ 𝐴 (𝐵 ∩ 𝐶))

Syntax hints:   → wi 4   = wceq 1533   ≠ wne 2932   ∩ cin 3940  ∅c0 4315  ∩ ciin 4989

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-12 2163  ax-ext 2695

This theorem depends on definitions:  df-bi 206  df-an 396  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-clab 2702  df-cleq 2716  df-clel 2802  df-ne 2933  df-ral 3054  df-v 3468  df-dif 3944  df-in 3948  df-nul 4316  df-iin 4991

This theorem is referenced by: (None)