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Theorem inin 29931
 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 4047 . 2 (𝐴 ∩ (𝐴𝐵)) = (𝐵 ∩ (𝐴𝐴))
2 inidm 4043 . . 3 (𝐴𝐴) = 𝐴
32ineq2i 4034 . 2 (𝐵 ∩ (𝐴𝐴)) = (𝐵𝐴)
4 incom 4028 . 2 (𝐵𝐴) = (𝐴𝐵)
51, 3, 43eqtri 2806 1 (𝐴 ∩ (𝐴𝐵)) = (𝐴𝐵)
 Colors of variables: wff setvar class Syntax hints:   = wceq 1601   ∩ cin 3791 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1839  ax-4 1853  ax-5 1953  ax-6 2021  ax-7 2055  ax-9 2116  ax-10 2135  ax-11 2150  ax-12 2163  ax-ext 2754 This theorem depends on definitions:  df-bi 199  df-an 387  df-or 837  df-tru 1605  df-ex 1824  df-nf 1828  df-sb 2012  df-clab 2764  df-cleq 2770  df-clel 2774  df-nfc 2921  df-v 3400  df-in 3799 This theorem is referenced by:  measinb2  30892
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