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	⊢ ( 𝜑 → 𝐴 ∈ ℂ )
	mvlraddd.2	⊢ ( 𝜑 → 𝐵 ∈ ℂ )
	mvlraddd.3	⊢ ( 𝜑 → ( 𝐴 + 𝐵 ) = 𝐶 )
Assertion	mvlladdd	⊢ ( 𝜑 → 𝐵 = ( 𝐶 − 𝐴 ) )

Proof

Step	Hyp	Ref	Expression
1		mvlraddd.1	⊢ ( 𝜑 → 𝐴 ∈ ℂ )
2		mvlraddd.2	⊢ ( 𝜑 → 𝐵 ∈ ℂ )
3		mvlraddd.3	⊢ ( 𝜑 → ( 𝐴 + 𝐵 ) = 𝐶 )
4	2 1	pncand	⊢ ( 𝜑 → ( ( 𝐵 + 𝐴 ) − 𝐴 ) = 𝐵 )
5	1 2	addcomd	⊢ ( 𝜑 → ( 𝐴 + 𝐵 ) = ( 𝐵 + 𝐴 ) )
6	5 3	eqtr3d	⊢ ( 𝜑 → ( 𝐵 + 𝐴 ) = 𝐶 )
7	6	oveq1d	⊢ ( 𝜑 → ( ( 𝐵 + 𝐴 ) − 𝐴 ) = ( 𝐶 − 𝐴 ) )
8	4 7	eqtr3d	⊢ ( 𝜑 → 𝐵 = ( 𝐶 − 𝐴 ) )