Theorem mvlladdi

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

	Ref	Expression
Hypotheses	mvlladdi.1	⊢ 𝐴 ∈ ℂ
	mvlladdi.2	⊢ 𝐵 ∈ ℂ
	mvlladdi.3	⊢ ( 𝐴 + 𝐵 ) = 𝐶
Assertion	mvlladdi	⊢ 𝐵 = ( 𝐶 − 𝐴 )

Proof

Step	Hyp	Ref	Expression
1		mvlladdi.1	⊢ 𝐴 ∈ ℂ
2		mvlladdi.2	⊢ 𝐵 ∈ ℂ
3		mvlladdi.3	⊢ ( 𝐴 + 𝐵 ) = 𝐶
4	2 1	pncan3oi	⊢ ( ( 𝐵 + 𝐴 ) − 𝐴 ) = 𝐵
5	1 2 3	addcomli	⊢ ( 𝐵 + 𝐴 ) = 𝐶
6	5	oveq1i	⊢ ( ( 𝐵 + 𝐴 ) − 𝐴 ) = ( 𝐶 − 𝐴 )
7	4 6	eqtr3i	⊢ 𝐵 = ( 𝐶 − 𝐴 )