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Metamath Proof Explorer

Theorem subadd4b

Description: Rearrangement of 4 terms in a mixed addition and subtraction. (Contributed by Glauco Siliprandi, 11-Dec-2019)

Ref Expression
Hypotheses subadd4b.1 
|- ( ph -> A e. CC )
subadd4b.2 
|- ( ph -> B e. CC )
subadd4b.3 
|- ( ph -> C e. CC )
subadd4b.4 
|- ( ph -> D e. CC )
Assertion subadd4b 
|- ( ph -> ( ( A - B ) + ( C - D ) ) = ( ( A - D ) + ( C - B ) ) )

Proof

Step Hyp Ref Expression
1 subadd4b.1 
 |-  ( ph -> A e. CC )
2 subadd4b.2 
 |-  ( ph -> B e. CC )
3 subadd4b.3 
 |-  ( ph -> C e. CC )
4 subadd4b.4 
 |-  ( ph -> D e. CC )
5 1 2 4 3 subadd4d 
 |-  ( ph -> ( ( A - B ) - ( D - C ) ) = ( ( A + C ) - ( B + D ) ) )
6 1 2 subcld 
 |-  ( ph -> ( A - B ) e. CC )
7 6 4 3 subsub2d 
 |-  ( ph -> ( ( A - B ) - ( D - C ) ) = ( ( A - B ) + ( C - D ) ) )
8 2 4 addcomd 
 |-  ( ph -> ( B + D ) = ( D + B ) )
9 8 oveq2d 
 |-  ( ph -> ( ( A + C ) - ( B + D ) ) = ( ( A + C ) - ( D + B ) ) )
10 1 3 4 2 addsub4d 
 |-  ( ph -> ( ( A + C ) - ( D + B ) ) = ( ( A - D ) + ( C - B ) ) )
11 9 10 eqtrd 
 |-  ( ph -> ( ( A + C ) - ( B + D ) ) = ( ( A - D ) + ( C - B ) ) )
12 5 7 11 3eqtr3d 
 |-  ( ph -> ( ( A - B ) + ( C - D ) ) = ( ( A - D ) + ( C - B ) ) )