Metamath Proof Explorer

Theorem dif32

Description: Swap second and third argument of double difference. (Contributed by NM, 18-Aug-2004)

Ref Expression
Assertion dif32 
|- ( ( A \ B ) \ C ) = ( ( A \ C ) \ B )

Proof

Step Hyp Ref Expression
1 uncom 
 |-  ( B u. C ) = ( C u. B )
2 1 difeq2i 
 |-  ( A \ ( B u. C ) ) = ( A \ ( C u. B ) )
3 difun1 
 |-  ( A \ ( B u. C ) ) = ( ( A \ B ) \ C )
4 difun1 
 |-  ( A \ ( C u. B ) ) = ( ( A \ C ) \ B )
5 2 3 4 3eqtr3i 
 |-  ( ( A \ B ) \ C ) = ( ( A \ C ) \ B )