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Theorem inabs3 44299
Description: Absorption law for intersection. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
Assertion
Ref Expression
inabs3 (𝐶𝐵 → ((𝐴𝐵) ∩ 𝐶) = (𝐴𝐶))

Proof of Theorem inabs3
StepHypRef Expression
1 inass 4214 . 2 ((𝐴𝐵) ∩ 𝐶) = (𝐴 ∩ (𝐵𝐶))
2 sseqin2 4210 . . . 4 (𝐶𝐵 ↔ (𝐵𝐶) = 𝐶)
32biimpi 215 . . 3 (𝐶𝐵 → (𝐵𝐶) = 𝐶)
43ineq2d 4207 . 2 (𝐶𝐵 → (𝐴 ∩ (𝐵𝐶)) = (𝐴𝐶))
51, 4eqtrid 2778 1 (𝐶𝐵 → ((𝐴𝐵) ∩ 𝐶) = (𝐴𝐶))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1533  cin 3942  wss 3943
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-ext 2697
This theorem depends on definitions:  df-bi 206  df-an 396  df-tru 1536  df-ex 1774  df-sb 2060  df-clab 2704  df-cleq 2718  df-clel 2804  df-rab 3427  df-v 3470  df-in 3950  df-ss 3960
This theorem is referenced by:  carageniuncllem1  45790
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