Theorem inabs3 45633

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 4179	. 2 ⊢ ((𝐴 ∩ 𝐵) ∩ 𝐶) = (𝐴 ∩ (𝐵 ∩ 𝐶))
2		sseqin2 4175	. . . 4 ⊢ (𝐶 ⊆ 𝐵 ↔ (𝐵 ∩ 𝐶) = 𝐶)
3	2	biimpi 218	. . 3 ⊢ (𝐶 ⊆ 𝐵 → (𝐵 ∩ 𝐶) = 𝐶)
4	3	ineq2d 4172	. 2 ⊢ (𝐶 ⊆ 𝐵 → (𝐴 ∩ (𝐵 ∩ 𝐶)) = (𝐴 ∩ 𝐶))
5	1, 4	eqtrid 2809	1 ⊢ (𝐶 ⊆ 𝐵 → ((𝐴 ∩ 𝐵) ∩ 𝐶) = (𝐴 ∩ 𝐶))

Syntax hints:   → wi 4   = wceq 1560   ∩ cin 3903   ⊆ wss 3904

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1815  ax-4 1829  ax-5 1930  ax-6 1987  ax-7 2028  ax-8 2144  ax-9 2152  ax-ext 2734

This theorem depends on definitions:  df-bi 209  df-an 400  df-3an 1100  df-tru 1563  df-ex 1800  df-sb 2091  df-clab 2741  df-cleq 2754  df-clel 2837  df-rab 3415  df-v 3456  df-in 3911  df-ss 3921

This theorem is referenced by:  carageniuncllem1  47092