Metamath Proof Explorer


Theorem mvllmuld

Description: Move the left term in a product on the LHS to the RHS, deduction form. (Contributed by David A. Wheeler, 11-Oct-2018)

Ref Expression
Hypotheses mvllmuld.1 φ A
mvllmuld.2 φ B
mvllmuld.3 φ A 0
mvllmuld.4 φ A B = C
Assertion mvllmuld φ B = C A

Proof

Step Hyp Ref Expression
1 mvllmuld.1 φ A
2 mvllmuld.2 φ B
3 mvllmuld.3 φ A 0
4 mvllmuld.4 φ A B = C
5 2 1 3 divcan4d φ B A A = B
6 1 2 mulcomd φ A B = B A
7 6 4 eqtr3d φ B A = C
8 7 oveq1d φ B A A = C A
9 5 8 eqtr3d φ B = C A