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Metamath Proof Explorer

Theorem subadd2i

Description: Relationship between subtraction and addition. (Contributed by NM, 15-Dec-2006)

Ref Expression
Hypotheses negidi.1 ⊢ 𝐴 ∈ ℂ
pncan3i.2 ⊢ 𝐵 ∈ ℂ
subadd.3 ⊢ 𝐶 ∈ ℂ
Assertion subadd2i ( ( 𝐴 − 𝐵 ) = 𝐶 ↔ ( 𝐶 + 𝐵 ) = 𝐴 )

Proof

Step Hyp Ref Expression
1 negidi.1 ⊢ 𝐴 ∈ ℂ
2 pncan3i.2 ⊢ 𝐵 ∈ ℂ
3 subadd.3 ⊢ 𝐶 ∈ ℂ
4 subadd2 ⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ ) → ( ( 𝐴 − 𝐵 ) = 𝐶 ↔ ( 𝐶 + 𝐵 ) = 𝐴 ) )
5 1 2 3 4 mp3an ⊢ ( ( 𝐴 − 𝐵 ) = 𝐶 ↔ ( 𝐶 + 𝐵 ) = 𝐴 )