Metamath Proof Explorer


Theorem subsub4

Description: Law for double subtraction. (Contributed by NM, 19-Aug-2005) (Revised by Mario Carneiro, 27-May-2016)

Ref Expression
Assertion subsub4 ABCA-B-C=AB+C

Proof

Step Hyp Ref Expression
1 nppcan2 ABCA-B+C+C=AB
2 simp1 ABCA
3 simp2 ABCB
4 subcl ABAB
5 2 3 4 syl2anc ABCAB
6 simp3 ABCC
7 3 6 addcld ABCB+C
8 subcl AB+CAB+C
9 2 7 8 syl2anc ABCAB+C
10 subadd2 ABCAB+CA-B-C=AB+CA-B+C+C=AB
11 5 6 9 10 syl3anc ABCA-B-C=AB+CA-B+C+C=AB
12 1 11 mpbird ABCA-B-C=AB+C