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| Description: Associative-type law for subtraction and addition. (Contributed by Mario Carneiro, 27-May-2016.) | 
| Ref | Expression | 
|---|---|
| negidd.1 | ⊢ (𝜑 → 𝐴 ∈ ℂ) | 
| pncand.2 | ⊢ (𝜑 → 𝐵 ∈ ℂ) | 
| subaddd.3 | ⊢ (𝜑 → 𝐶 ∈ ℂ) | 
| Ref | Expression | 
|---|---|
| addsubassd | ⊢ (𝜑 → ((𝐴 + 𝐵) − 𝐶) = (𝐴 + (𝐵 − 𝐶))) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | negidd.1 | . 2 ⊢ (𝜑 → 𝐴 ∈ ℂ) | |
| 2 | pncand.2 | . 2 ⊢ (𝜑 → 𝐵 ∈ ℂ) | |
| 3 | subaddd.3 | . 2 ⊢ (𝜑 → 𝐶 ∈ ℂ) | |
| 4 | addsubass 11518 | . 2 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ) → ((𝐴 + 𝐵) − 𝐶) = (𝐴 + (𝐵 − 𝐶))) | |
| 5 | 1, 2, 3, 4 | syl3anc 1373 | 1 ⊢ (𝜑 → ((𝐴 + 𝐵) − 𝐶) = (𝐴 + (𝐵 − 𝐶))) | 
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