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Metamath Proof Explorer

Theorem subaddrii

Description: Relationship between subtraction and addition. (Contributed by NM, 16-Dec-2006)

Ref Expression
Hypotheses negidi.1 ⊢ 𝐴 ∈ ℂ
pncan3i.2 ⊢ 𝐵 ∈ ℂ
subadd.3 ⊢ 𝐶 ∈ ℂ
subaddri.4 ⊢ ( 𝐵 + 𝐶 ) = 𝐴
Assertion subaddrii ( 𝐴 − 𝐵 ) = 𝐶

Proof

Step Hyp Ref Expression
1 negidi.1 ⊢ 𝐴 ∈ ℂ
2 pncan3i.2 ⊢ 𝐵 ∈ ℂ
3 subadd.3 ⊢ 𝐶 ∈ ℂ
4 subaddri.4 ⊢ ( 𝐵 + 𝐶 ) = 𝐴
5 1 2 3 subaddi ⊢ ( ( 𝐴 − 𝐵 ) = 𝐶 ↔ ( 𝐵 + 𝐶 ) = 𝐴 )
6 4 5 mpbir ⊢ ( 𝐴 − 𝐵 ) = 𝐶