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Theorem inin 30263
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 4174 . 2 (𝐴 ∩ (𝐴𝐵)) = (𝐵 ∩ (𝐴𝐴))
2 inidm 4170 . . 3 (𝐴𝐴) = 𝐴
32ineq2i 4161 . 2 (𝐵 ∩ (𝐴𝐴)) = (𝐵𝐴)
4 incom 4153 . 2 (𝐵𝐴) = (𝐴𝐵)
51, 3, 43eqtri 2848 1 (𝐴 ∩ (𝐴𝐵)) = (𝐴𝐵)
Colors of variables: wff setvar class
Syntax hints:   = wceq 1538  cin 3909
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-ext 2793
This theorem depends on definitions:  df-bi 210  df-an 400  df-tru 1541  df-ex 1782  df-sb 2071  df-clab 2800  df-cleq 2814  df-clel 2892  df-rab 3135  df-v 3473  df-in 3917
This theorem is referenced by:  measinb2  31489
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