Theorem inabs3 38249

Description: Absorption law for intersection. (Contributed by Glauco Siliprandi, 17-Aug-2020.)

Assertion

Ref	Expression
inabs3	⊢ (𝐶 ⊆ 𝐵 → ((𝐴 ∩ 𝐵) ∩ 𝐶) = (𝐴 ∩ 𝐶))

Proof of Theorem inabs3

Step	Hyp	Ref	Expression
1		inass 3785	. . 3 ⊢ ((𝐴 ∩ 𝐵) ∩ 𝐶) = (𝐴 ∩ (𝐵 ∩ 𝐶))
2	1	a1i 11	. 2 ⊢ (𝐶 ⊆ 𝐵 → ((𝐴 ∩ 𝐵) ∩ 𝐶) = (𝐴 ∩ (𝐵 ∩ 𝐶)))
3		sseqin2 3779	. . . 4 ⊢ (𝐶 ⊆ 𝐵 ↔ (𝐵 ∩ 𝐶) = 𝐶)
4	3	biimpi 205	. . 3 ⊢ (𝐶 ⊆ 𝐵 → (𝐵 ∩ 𝐶) = 𝐶)
5	4	ineq2d 3776	. 2 ⊢ (𝐶 ⊆ 𝐵 → (𝐴 ∩ (𝐵 ∩ 𝐶)) = (𝐴 ∩ 𝐶))
6		eqidd 2611	. 2 ⊢ (𝐶 ⊆ 𝐵 → (𝐴 ∩ 𝐶) = (𝐴 ∩ 𝐶))
7	2, 5, 6	3eqtrd 2648	1 ⊢ (𝐶 ⊆ 𝐵 → ((𝐴 ∩ 𝐵) ∩ 𝐶) = (𝐴 ∩ 𝐶))

Syntax hints:   → wi 4   = wceq 1475   ∩ cin 3539   ⊆ wss 3540

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-v 3175  df-in 3547  df-ss 3554

This theorem is referenced by:  carageniuncllem1  39411