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Theorem inin 28737
 Description: Intersection with an intersection. (Contributed by Thierry Arnoux, 27-Dec-2016.)
Assertion
Ref Expression
inin (𝐴 ∩ (𝐴𝐵)) = (𝐴𝐵)

Proof of Theorem inin
StepHypRef Expression
1 in13 3788 . 2 (𝐴 ∩ (𝐴𝐵)) = (𝐵 ∩ (𝐴𝐴))
2 inidm 3784 . . 3 (𝐴𝐴) = 𝐴
32ineq2i 3773 . 2 (𝐵 ∩ (𝐴𝐴)) = (𝐵𝐴)
4 incom 3767 . 2 (𝐵𝐴) = (𝐴𝐵)
51, 3, 43eqtri 2636 1 (𝐴 ∩ (𝐴𝐵)) = (𝐴𝐵)
 Colors of variables: wff setvar class Syntax hints:   = wceq 1475   ∩ cin 3539 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 This theorem is referenced by:  measinb2  29613
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