Metamath Proof Explorer


Theorem npcand

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

Ref Expression
Hypotheses negidd.1 ( 𝜑𝐴 ∈ ℂ )
pncand.2 ( 𝜑𝐵 ∈ ℂ )
Assertion npcand ( 𝜑 → ( ( 𝐴𝐵 ) + 𝐵 ) = 𝐴 )

Proof

Step Hyp Ref Expression
1 negidd.1 ( 𝜑𝐴 ∈ ℂ )
2 pncand.2 ( 𝜑𝐵 ∈ ℂ )
3 npcan ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ) → ( ( 𝐴𝐵 ) + 𝐵 ) = 𝐴 )
4 1 2 3 syl2anc ( 𝜑 → ( ( 𝐴𝐵 ) + 𝐵 ) = 𝐴 )