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Metamath Proof Explorer

Theorem pncan

Description: Cancellation law for subtraction. (Contributed by NM, 10-May-2004) (Revised by Mario Carneiro, 27-May-2016)

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

Proof

Step Hyp Ref Expression
1 simpr ⊢ A ∈ ℂ ∧ B ∈ ℂ → B ∈ ℂ
2 simpl ⊢ A ∈ ℂ ∧ B ∈ ℂ → A ∈ ℂ
3 1 2 addcomd ⊢ A ∈ ℂ ∧ B ∈ ℂ → B + A = A + B
4 addcl ⊢ A ∈ ℂ ∧ B ∈ ℂ → A + B ∈ ℂ
5 subadd ⊢ A + B ∈ ℂ ∧ B ∈ ℂ ∧ A ∈ ℂ → A + B - B = A ↔ B + A = A + B
6 4 1 2 5 syl3anc ⊢ A ∈ ℂ ∧ B ∈ ℂ → A + B - B = A ↔ B + A = A + B
7 3 6 mpbird ⊢ A ∈ ℂ ∧ B ∈ ℂ → A + B - B = A