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Metamath Proof Explorer

Theorem addcan2i

Description: Cancellation law for addition. Theorem I.1 of Apostol p. 18. (Contributed by NM, 14-May-2003) (Revised by Scott Fenton, 3-Jan-2013)

Ref Expression
Hypotheses mul.1 ⊢ 𝐴 ∈ ℂ
mul.2 ⊢ 𝐵 ∈ ℂ
mul.3 ⊢ 𝐶 ∈ ℂ
Assertion addcan2i ( ( 𝐴 + 𝐶 ) = ( 𝐵 + 𝐶 ) ↔ 𝐴 = 𝐵 )

Proof

Step Hyp Ref Expression
1 mul.1 ⊢ 𝐴 ∈ ℂ
2 mul.2 ⊢ 𝐵 ∈ ℂ
3 mul.3 ⊢ 𝐶 ∈ ℂ
4 addcan2 ⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ ) → ( ( 𝐴 + 𝐶 ) = ( 𝐵 + 𝐶 ) ↔ 𝐴 = 𝐵 ) )
5 1 2 3 4 mp3an ⊢ ( ( 𝐴 + 𝐶 ) = ( 𝐵 + 𝐶 ) ↔ 𝐴 = 𝐵 )