Metamath Proof Explorer


Theorem sub4d

Description: Rearrangement of 4 terms in a subtraction. (Contributed by Mario Carneiro, 27-May-2016)

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 sub4d
|- ( 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 sub4
 |-  ( ( ( A e. CC /\ B e. CC ) /\ ( C e. CC /\ D e. CC ) ) -> ( ( A - B ) - ( C - D ) ) = ( ( A - C ) - ( B - D ) ) )
6 1 2 3 4 5 syl22anc
 |-  ( ph -> ( ( A - B ) - ( C - D ) ) = ( ( A - C ) - ( B - D ) ) )