Metamath Proof Explorer


Theorem div12d

Description: A commutative/associative law for division. (Contributed by Mario Carneiro, 27-May-2016)

Ref Expression
Hypotheses div1d.1 φ A
divcld.2 φ B
divmuld.3 φ C
divassd.4 φ C 0
Assertion div12d φ A B C = B A C

Proof

Step Hyp Ref Expression
1 div1d.1 φ A
2 divcld.2 φ B
3 divmuld.3 φ C
4 divassd.4 φ C 0
5 div12 A B C C 0 A B C = B A C
6 1 2 3 4 5 syl112anc φ A B C = B A C