Metamath Proof Explorer


Theorem negsubdid

Description: Distribution of negative over subtraction. (Contributed by Mario Carneiro, 27-May-2016)

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

Proof

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