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Metamath Proof Explorer

Theorem pnpcan

Description: Cancellation law for mixed addition and subtraction. (Contributed by NM, 4-Mar-2005) (Revised by Mario Carneiro, 27-May-2016) (Proof shortened by SN, 13-Nov-2023)

Ref Expression
Assertion pnpcan ⊢ A ∈ ℂ ∧ B ∈ ℂ ∧ C ∈ ℂ → A + B - A + C = B − C

Proof

Step Hyp Ref Expression
1 addcl ⊢ A ∈ ℂ ∧ B ∈ ℂ → A + B ∈ ℂ
2 subsub4 ⊢ A + B ∈ ℂ ∧ A ∈ ℂ ∧ C ∈ ℂ → A + B - A - C = A + B - A + C
3 1 2 stoic4a ⊢ A ∈ ℂ ∧ B ∈ ℂ ∧ C ∈ ℂ → A + B - A - C = A + B - A + C
4 pncan2 ⊢ A ∈ ℂ ∧ B ∈ ℂ → A + B - A = B
5 4 3adant3 ⊢ A ∈ ℂ ∧ B ∈ ℂ ∧ C ∈ ℂ → A + B - A = B
6 5 oveq1d ⊢ A ∈ ℂ ∧ B ∈ ℂ ∧ C ∈ ℂ → A + B - A - C = B − C
7 3 6 eqtr3d ⊢ A ∈ ℂ ∧ B ∈ ℂ ∧ C ∈ ℂ → A + B - A + C = B − C