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Theorem inabs3 38249
 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 3785 . . 3 ((𝐴𝐵) ∩ 𝐶) = (𝐴 ∩ (𝐵𝐶))
21a1i 11 . 2 (𝐶𝐵 → ((𝐴𝐵) ∩ 𝐶) = (𝐴 ∩ (𝐵𝐶)))
3 sseqin2 3779 . . . 4 (𝐶𝐵 ↔ (𝐵𝐶) = 𝐶)
43biimpi 205 . . 3 (𝐶𝐵 → (𝐵𝐶) = 𝐶)
54ineq2d 3776 . 2 (𝐶𝐵 → (𝐴 ∩ (𝐵𝐶)) = (𝐴𝐶))
6 eqidd 2611 . 2 (𝐶𝐵 → (𝐴𝐶) = (𝐴𝐶))
72, 5, 63eqtrd 2648 1 (𝐶𝐵 → ((𝐴𝐵) ∩ 𝐶) = (𝐴𝐶))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   = wceq 1475   ∩ cin 3539   ⊆ wss 3540 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-v 3175  df-in 3547  df-ss 3554 This theorem is referenced by:  carageniuncllem1  39411
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