Theorem iinin1m 3829

Description: Indexed intersection of intersection. Compare to Theorem "Distributive laws" in [Enderton] p. 30. (Contributed by Jim Kingdon, 17-Aug-2018.)

Assertion

Ref	Expression
iinin1m	⊢ (∃𝑥 𝑥 ∈ 𝐴 → ∩ 𝑥 ∈ 𝐴 (𝐶 ∩ 𝐵) = (∩ 𝑥 ∈ 𝐴 𝐶 ∩ 𝐵))

Distinct variable groups:   𝑥,𝐴   𝑥,𝐵

Allowed substitution hint:   𝐶(𝑥)

Proof of Theorem iinin1m

Step	Hyp	Ref	Expression
1		iinin2m 3828	. 2 ⊢ (∃𝑥 𝑥 ∈ 𝐴 → ∩ 𝑥 ∈ 𝐴 (𝐵 ∩ 𝐶) = (𝐵 ∩ ∩ 𝑥 ∈ 𝐴 𝐶))
2		incom 3215	. . . 4 ⊢ (𝐶 ∩ 𝐵) = (𝐵 ∩ 𝐶)
3	2	a1i 9	. . 3 ⊢ (𝑥 ∈ 𝐴 → (𝐶 ∩ 𝐵) = (𝐵 ∩ 𝐶))
4	3	iineq2i 3779	. 2 ⊢ ∩ 𝑥 ∈ 𝐴 (𝐶 ∩ 𝐵) = ∩ 𝑥 ∈ 𝐴 (𝐵 ∩ 𝐶)
5		incom 3215	. 2 ⊢ (∩ 𝑥 ∈ 𝐴 𝐶 ∩ 𝐵) = (𝐵 ∩ ∩ 𝑥 ∈ 𝐴 𝐶)
6	1, 4, 5	3eqtr4g 2157	1 ⊢ (∃𝑥 𝑥 ∈ 𝐴 → ∩ 𝑥 ∈ 𝐴 (𝐶 ∩ 𝐵) = (∩ 𝑥 ∈ 𝐴 𝐶 ∩ 𝐵))

Syntax hints:   → wi 4   = wceq 1299  ∃wex 1436   ∈ wcel 1448   ∩ cin 3020  ∩ ciin 3761

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 671  ax-5 1391  ax-7 1392  ax-gen 1393  ax-ie1 1437  ax-ie2 1438  ax-8 1450  ax-10 1451  ax-11 1452  ax-i12 1453  ax-bndl 1454  ax-4 1455  ax-17 1474  ax-i9 1478  ax-ial 1482  ax-i5r 1483  ax-ext 2082

This theorem depends on definitions:  df-bi 116  df-tru 1302  df-nf 1405  df-sb 1704  df-clab 2087  df-cleq 2093  df-clel 2096  df-nfc 2229  df-ral 2380  df-v 2643  df-in 3027  df-iin 3763

This theorem is referenced by: (None)