Theorem inabs3 45050

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 4191	. 2 ⊢ ((𝐴 ∩ 𝐵) ∩ 𝐶) = (𝐴 ∩ (𝐵 ∩ 𝐶))
2		sseqin2 4186	. . . 4 ⊢ (𝐶 ⊆ 𝐵 ↔ (𝐵 ∩ 𝐶) = 𝐶)
3	2	biimpi 216	. . 3 ⊢ (𝐶 ⊆ 𝐵 → (𝐵 ∩ 𝐶) = 𝐶)
4	3	ineq2d 4183	. 2 ⊢ (𝐶 ⊆ 𝐵 → (𝐴 ∩ (𝐵 ∩ 𝐶)) = (𝐴 ∩ 𝐶))
5	1, 4	eqtrid 2776	1 ⊢ (𝐶 ⊆ 𝐵 → ((𝐴 ∩ 𝐵) ∩ 𝐶) = (𝐴 ∩ 𝐶))

Syntax hints:   → wi 4   = wceq 1540   ∩ cin 3913   ⊆ wss 3914

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-ext 2701

This theorem depends on definitions:  df-bi 207  df-an 396  df-3an 1088  df-tru 1543  df-ex 1780  df-sb 2066  df-clab 2708  df-cleq 2721  df-clel 2803  df-rab 3406  df-v 3449  df-in 3921  df-ss 3931

This theorem is referenced by:  carageniuncllem1  46519