Metamath Proof Explorer


Theorem mvrraddd

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

Ref Expression
Hypotheses mvrraddd.1 ( 𝜑𝐵 ∈ ℂ )
mvrraddd.2 ( 𝜑𝐶 ∈ ℂ )
mvrraddd.3 ( 𝜑𝐴 = ( 𝐵 + 𝐶 ) )
Assertion mvrraddd ( 𝜑 → ( 𝐴𝐶 ) = 𝐵 )

Proof

Step Hyp Ref Expression
1 mvrraddd.1 ( 𝜑𝐵 ∈ ℂ )
2 mvrraddd.2 ( 𝜑𝐶 ∈ ℂ )
3 mvrraddd.3 ( 𝜑𝐴 = ( 𝐵 + 𝐶 ) )
4 3 oveq1d ( 𝜑 → ( 𝐴𝐶 ) = ( ( 𝐵 + 𝐶 ) − 𝐶 ) )
5 1 2 pncand ( 𝜑 → ( ( 𝐵 + 𝐶 ) − 𝐶 ) = 𝐵 )
6 4 5 eqtrd ( 𝜑 → ( 𝐴𝐶 ) = 𝐵 )