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Theorem inabs3 39719
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 3962 . 2 ((𝐴𝐵) ∩ 𝐶) = (𝐴 ∩ (𝐵𝐶))
2 sseqin2 3956 . . . 4 (𝐶𝐵 ↔ (𝐵𝐶) = 𝐶)
32biimpi 206 . . 3 (𝐶𝐵 → (𝐵𝐶) = 𝐶)
43ineq2d 3953 . 2 (𝐶𝐵 → (𝐴 ∩ (𝐵𝐶)) = (𝐴𝐶))
51, 4syl5eq 2802 1 (𝐶𝐵 → ((𝐴𝐵) ∩ 𝐶) = (𝐴𝐶))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1628  cin 3710  wss 3711
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1867  ax-4 1882  ax-5 1984  ax-6 2050  ax-7 2086  ax-9 2144  ax-10 2164  ax-11 2179  ax-12 2192  ax-13 2387  ax-ext 2736
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-tru 1631  df-ex 1850  df-nf 1855  df-sb 2043  df-clab 2743  df-cleq 2749  df-clel 2752  df-nfc 2887  df-v 3338  df-in 3718  df-ss 3725
This theorem is referenced by:  carageniuncllem1  41237
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