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Theorem inin 30204
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 4196 . 2 (𝐴 ∩ (𝐴𝐵)) = (𝐵 ∩ (𝐴𝐴))
2 inidm 4192 . . 3 (𝐴𝐴) = 𝐴
32ineq2i 4183 . 2 (𝐵 ∩ (𝐴𝐴)) = (𝐵𝐴)
4 incom 4175 . 2 (𝐵𝐴) = (𝐴𝐵)
51, 3, 43eqtri 2845 1 (𝐴 ∩ (𝐴𝐵)) = (𝐴𝐵)
Colors of variables: wff setvar class
Syntax hints:   = wceq 1528  cin 3932
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-rab 3144  df-v 3494  df-in 3940
This theorem is referenced by:  measinb2  31381
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