Theorem iniin1 46083

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

Assertion

Ref	Expression
iniin1	⊢ (𝐴 ≠ ∅ → (∩ 𝑥 ∈ 𝐴 𝐶 ∩ 𝐵) = ∩ 𝑥 ∈ 𝐴 (𝐶 ∩ 𝐵))

Distinct variable groups:   𝑥,𝐴   𝑥,𝐵

Allowed substitution hint:   𝐶(𝑥)

Proof of Theorem iniin1

Step	Hyp	Ref	Expression
1		iinin1 5039	. 2 ⊢ (𝐴 ≠ ∅ → ∩ 𝑥 ∈ 𝐴 (𝐶 ∩ 𝐵) = (∩ 𝑥 ∈ 𝐴 𝐶 ∩ 𝐵))
2	1	eqcomd 2767	1 ⊢ (𝐴 ≠ ∅ → (∩ 𝑥 ∈ 𝐴 𝐶 ∩ 𝐵) = ∩ 𝑥 ∈ 𝐴 (𝐶 ∩ 𝐵))

This proof depends on syntax axioms:   → wi 4   = wceq 1570   ≠ wne 2956   ∩ cin 3898  ∅c0 4279  ∩ ciin 4952

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-12 2213  ax-ext 2733

This proof depends on definitions:  df-bi 210  df-an 402  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-clab 2740  df-cleq 2753  df-clel 2836  df-ne 2957  df-ral 3078  df-rab 3414  df-v 3453  df-dif 3902  df-in 3906  df-nul 4280  df-iin 4954

This theorem is used by:  smfsuplem1  47765