Metamath Proof Explorer


Theorem pnncan

Description: Cancellation law for mixed addition and subtraction. (Contributed by NM, 30-Jun-2005) (Revised by Mario Carneiro, 27-May-2016)

Ref Expression
Assertion pnncan A B C A + B - A C = B + C

Proof

Step Hyp Ref Expression
1 simp1 A B C A
2 simp2 A B C B
3 1 2 addcld A B C A + B
4 simp3 A B C C
5 subsub A + B A C A + B - A C = A + B - A + C
6 3 1 4 5 syl3anc A B C A + B - A C = A + B - A + C
7 pncan2 A B A + B - A = B
8 7 3adant3 A B C A + B - A = B
9 8 oveq1d A B C A + B - A + C = B + C
10 6 9 eqtrd A B C A + B - A C = B + C