Theorem iniin2 45668

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 5034	. 2 ⊢ (𝐴 ≠ ∅ → ∩ 𝑥 ∈ 𝐴 (𝐵 ∩ 𝐶) = (𝐵 ∩ ∩ 𝑥 ∈ 𝐴 𝐶))
2	1	eqcomd 2767	1 ⊢ (𝐴 ≠ ∅ → (𝐵 ∩ ∩ 𝑥 ∈ 𝐴 𝐶) = ∩ 𝑥 ∈ 𝐴 (𝐵 ∩ 𝐶))

Syntax hints:   → wi 4   = wceq 1559   ≠ wne 2956   ∩ cin 3903  ∅c0 4285  ∩ ciin 4949

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1814  ax-4 1828  ax-5 1929  ax-6 1986  ax-7 2027  ax-8 2143  ax-9 2151  ax-12 2211  ax-ext 2733

This theorem depends on definitions:  df-bi 209  df-an 400  df-tru 1562  df-fal 1572  df-ex 1799  df-nf 1803  df-sb 2090  df-clab 2740  df-cleq 2753  df-clel 2836  df-ne 2957  df-ral 3076  df-v 3455  df-dif 3907  df-in 3911  df-nul 4286  df-iin 4951

This theorem is referenced by: (None)