Metamath Proof Explorer


Theorem ppncand

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

Ref Expression
Hypotheses negidd.1 φ A
pncand.2 φ B
subaddd.3 φ C
Assertion ppncand φ A + B + C B = A + C

Proof

Step Hyp Ref Expression
1 negidd.1 φ A
2 pncand.2 φ B
3 subaddd.3 φ C
4 ppncan A B C A + B + C B = A + C
5 1 2 3 4 syl3anc φ A + B + C B = A + C