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Theorem inabs3 39741
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 3966 . 2 ((𝐴𝐵) ∩ 𝐶) = (𝐴 ∩ (𝐵𝐶))
2 sseqin2 3960 . . . 4 (𝐶𝐵 ↔ (𝐵𝐶) = 𝐶)
32biimpi 206 . . 3 (𝐶𝐵 → (𝐵𝐶) = 𝐶)
43ineq2d 3957 . 2 (𝐶𝐵 → (𝐴 ∩ (𝐵𝐶)) = (𝐴𝐶))
51, 4syl5eq 2806 1 (𝐶𝐵 → ((𝐴𝐵) ∩ 𝐶) = (𝐴𝐶))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1632  cin 3714  wss 3715
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-v 3342  df-in 3722  df-ss 3729
This theorem is referenced by:  carageniuncllem1  41259
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