Metamath Proof Explorer


Theorem negsubdi3d

Description: Distribution of negative over subtraction. (Contributed by Glauco Siliprandi, 11-Dec-2019)

Ref Expression
Hypotheses negsubdi3d.1 ( 𝜑𝐴 ∈ ℂ )
negsubdi3d.2 ( 𝜑𝐵 ∈ ℂ )
Assertion negsubdi3d ( 𝜑 → - ( 𝐴𝐵 ) = ( - 𝐴 − - 𝐵 ) )

Proof

Step Hyp Ref Expression
1 negsubdi3d.1 ( 𝜑𝐴 ∈ ℂ )
2 negsubdi3d.2 ( 𝜑𝐵 ∈ ℂ )
3 1 2 negsubdi2d ( 𝜑 → - ( 𝐴𝐵 ) = ( 𝐵𝐴 ) )
4 1 2 neg2subd ( 𝜑 → ( - 𝐴 − - 𝐵 ) = ( 𝐵𝐴 ) )
5 3 4 eqtr4d ( 𝜑 → - ( 𝐴𝐵 ) = ( - 𝐴 − - 𝐵 ) )