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Theorem inabs3 44996
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 4236 . 2 ((𝐴𝐵) ∩ 𝐶) = (𝐴 ∩ (𝐵𝐶))
2 sseqin2 4231 . . . 4 (𝐶𝐵 ↔ (𝐵𝐶) = 𝐶)
32biimpi 216 . . 3 (𝐶𝐵 → (𝐵𝐶) = 𝐶)
43ineq2d 4228 . 2 (𝐶𝐵 → (𝐴 ∩ (𝐵𝐶)) = (𝐴𝐶))
51, 4eqtrid 2787 1 (𝐶𝐵 → ((𝐴𝐵) ∩ 𝐶) = (𝐴𝐶))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1537  cin 3962  wss 3963
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-ext 2706
This theorem depends on definitions:  df-bi 207  df-an 396  df-3an 1088  df-tru 1540  df-ex 1777  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-rab 3434  df-v 3480  df-in 3970  df-ss 3980
This theorem is referenced by:  carageniuncllem1  46477
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