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Metamath Proof Explorer

Theorem indifundif

Description: A remarkable equation with sets. (Contributed by Thierry Arnoux, 18-May-2020)

Ref Expression
Assertion indifundif 
|- ( ( ( A i^i B ) \ C ) u. ( A \ B ) ) = ( A \ ( B i^i C ) )

Proof

Step Hyp Ref Expression
1 difindi 
 |-  ( A \ ( B i^i C ) ) = ( ( A \ B ) u. ( A \ C ) )
2 difundir 
 |-  ( ( ( A i^i B ) u. ( A \ B ) ) \ C ) = ( ( ( A i^i B ) \ C ) u. ( ( A \ B ) \ C ) )
3 inundif 
 |-  ( ( A i^i B ) u. ( A \ B ) ) = A
4 3 difeq1i 
 |-  ( ( ( A i^i B ) u. ( A \ B ) ) \ C ) = ( A \ C )
5 uncom 
 |-  ( ( ( A i^i B ) \ C ) u. ( ( A \ B ) \ C ) ) = ( ( ( A \ B ) \ C ) u. ( ( A i^i B ) \ C ) )
6 2 4 5 3eqtr3i 
 |-  ( A \ C ) = ( ( ( A \ B ) \ C ) u. ( ( A i^i B ) \ C ) )
7 6 uneq2i 
 |-  ( ( A \ B ) u. ( A \ C ) ) = ( ( A \ B ) u. ( ( ( A \ B ) \ C ) u. ( ( A i^i B ) \ C ) ) )
8 unass 
 |-  ( ( ( A \ B ) u. ( ( A \ B ) \ C ) ) u. ( ( A i^i B ) \ C ) ) = ( ( A \ B ) u. ( ( ( A \ B ) \ C ) u. ( ( A i^i B ) \ C ) ) )
9 undifabs 
 |-  ( ( A \ B ) u. ( ( A \ B ) \ C ) ) = ( A \ B )
10 9 uneq1i 
 |-  ( ( ( A \ B ) u. ( ( A \ B ) \ C ) ) u. ( ( A i^i B ) \ C ) ) = ( ( A \ B ) u. ( ( A i^i B ) \ C ) )
11 7 8 10 3eqtr2i 
 |-  ( ( A \ B ) u. ( A \ C ) ) = ( ( A \ B ) u. ( ( A i^i B ) \ C ) )
12 uncom 
 |-  ( ( A \ B ) u. ( ( A i^i B ) \ C ) ) = ( ( ( A i^i B ) \ C ) u. ( A \ B ) )
13 1 11 12 3eqtrri 
 |-  ( ( ( A i^i B ) \ C ) u. ( A \ B ) ) = ( A \ ( B i^i C ) )