Theorem inabs3 46181

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 4123	. 2 ⊢ ((𝐴 ∩ 𝐵) ∩ 𝐶) = (𝐴 ∩ (𝐵 ∩ 𝐶))
2		sseqin2 4119	. . . 4 ⊢ (𝐶 ⊆ 𝐵 ↔ (𝐵 ∩ 𝐶) = 𝐶)
3	2	biimpi 219	. . 3 ⊢ (𝐶 ⊆ 𝐵 → (𝐵 ∩ 𝐶) = 𝐶)
4	3	ineq2d 4116	. 2 ⊢ (𝐶 ⊆ 𝐵 → (𝐴 ∩ (𝐵 ∩ 𝐶)) = (𝐴 ∩ 𝐶))
5	1, 4	eqtrid 2807	1 ⊢ (𝐶 ⊆ 𝐵 → ((𝐴 ∩ 𝐵) ∩ 𝐶) = (𝐴 ∩ 𝐶))

This proof depends on syntax axioms:   → wi 4   = wceq 1569   ⊆ wss 3896   ∩ cin 4088

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-ext 2732

This proof depends on definitions:  df-bi 210  df-an 402  df-3an 1105  df-tru 1572  df-ex 1812  df-sb 2099  df-clab 2739  df-cleq 2752  df-clel 2835  df-rab 3413  df-v 3452  df-ss 3897  df-in 4090

This theorem is used by:  carageniuncllem1  47639