Metamath Proof Explorer


Theorem pnncani

Description: Cancellation law for mixed addition and subtraction. (Contributed by NM, 14-Jan-2006)

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

Proof

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