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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 ⊢ φ → A ∈ ℂ
pncand.2 ⊢ φ → B ∈ ℂ
subaddd.3 ⊢ φ → C ∈ ℂ
addsub4d.4 ⊢ φ → D ∈ ℂ
Assertion subsubadd23 ⊢ φ → A - B - C + D = A - C - B + D

Proof

Step Hyp Ref Expression
1 negidd.1 ⊢ φ → A ∈ ℂ
2 pncand.2 ⊢ φ → B ∈ ℂ
3 subaddd.3 ⊢ φ → C ∈ ℂ
4 addsub4d.4 ⊢ φ → D ∈ ℂ
5 1 2 3 sub32d ⊢ φ → A - B - C = A - C - B
6 5 oveq1d ⊢ φ → A − B - C - D = A − C - B - D
7 1 2 subcld ⊢ φ → A − B ∈ ℂ
8 7 3 4 subsub4d ⊢ φ → A − B - C - D = A - B - C + D
9 1 3 subcld ⊢ φ → A − C ∈ ℂ
10 9 2 4 subsub4d ⊢ φ → A − C - B - D = A - C - B + D
11 6 8 10 3eqtr3d ⊢ φ → A - B - C + D = A - C - B + D