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Metamath Proof Explorer

Theorem subsubadd23

Description: Swap the second and the third terms in a difference of a difference and a sum. (Contributed by AV, 15-Nov-2025)

Ref Expression
Hypotheses negidd.1 
|- ( ph -> A e. CC )
pncand.2 
|- ( ph -> B e. CC )
subaddd.3 
|- ( ph -> C e. CC )
addsub4d.4 
|- ( ph -> D e. CC )
Assertion subsubadd23 
|- ( ph -> ( ( A - B ) - ( C + D ) ) = ( ( A - C ) - ( B + D ) ) )

Proof

Step Hyp Ref Expression
1 negidd.1 
 |-  ( ph -> A e. CC )
2 pncand.2 
 |-  ( ph -> B e. CC )
3 subaddd.3 
 |-  ( ph -> C e. CC )
4 addsub4d.4 
 |-  ( ph -> D e. CC )
5 1 2 3 sub32d 
 |-  ( ph -> ( ( A - B ) - C ) = ( ( A - C ) - B ) )
6 5 oveq1d 
 |-  ( ph -> ( ( ( A - B ) - C ) - D ) = ( ( ( A - C ) - B ) - D ) )
7 1 2 subcld 
 |-  ( ph -> ( A - B ) e. CC )
8 7 3 4 subsub4d 
 |-  ( ph -> ( ( ( A - B ) - C ) - D ) = ( ( A - B ) - ( C + D ) ) )
9 1 3 subcld 
 |-  ( ph -> ( A - C ) e. CC )
10 9 2 4 subsub4d 
 |-  ( ph -> ( ( ( A - C ) - B ) - D ) = ( ( A - C ) - ( B + D ) ) )
11 6 8 10 3eqtr3d 
 |-  ( ph -> ( ( A - B ) - ( C + D ) ) = ( ( A - C ) - ( B + D ) ) )