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Theorem inin 23949
Description: Intersection with an intersection (Contributed by Thierry Arnoux, 27-Dec-2016.)
Assertion
Ref Expression
inin  |-  ( A  i^i  ( A  i^i  B ) )  =  ( A  i^i  B )

Proof of Theorem inin
StepHypRef Expression
1 in13 3514 . 2  |-  ( A  i^i  ( A  i^i  B ) )  =  ( B  i^i  ( A  i^i  A ) )
2 inidm 3510 . . 3  |-  ( A  i^i  A )  =  A
32ineq2i 3499 . 2  |-  ( B  i^i  ( A  i^i  A ) )  =  ( B  i^i  A )
4 incom 3493 . 2  |-  ( B  i^i  A )  =  ( A  i^i  B
)
51, 3, 43eqtri 2428 1  |-  ( A  i^i  ( A  i^i  B ) )  =  ( A  i^i  B )
Colors of variables: wff set class
Syntax hints:    = wceq 1649    i^i cin 3279
This theorem is referenced by:  measinb2  24530
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385
This theorem depends on definitions:  df-bi 178  df-an 361  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-v 2918  df-in 3287
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