Metamath Proof Explorer


Theorem mvlladdd

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

Ref Expression
Hypotheses mvlraddd.1
|- ( ph -> A e. CC )
mvlraddd.2
|- ( ph -> B e. CC )
mvlraddd.3
|- ( ph -> ( A + B ) = C )
Assertion mvlladdd
|- ( ph -> B = ( C - A ) )

Proof

Step Hyp Ref Expression
1 mvlraddd.1
 |-  ( ph -> A e. CC )
2 mvlraddd.2
 |-  ( ph -> B e. CC )
3 mvlraddd.3
 |-  ( ph -> ( A + B ) = C )
4 2 1 pncand
 |-  ( ph -> ( ( B + A ) - A ) = B )
5 1 2 addcomd
 |-  ( ph -> ( A + B ) = ( B + A ) )
6 5 3 eqtr3d
 |-  ( ph -> ( B + A ) = C )
7 6 oveq1d
 |-  ( ph -> ( ( B + A ) - A ) = ( C - A ) )
8 4 7 eqtr3d
 |-  ( ph -> B = ( C - A ) )