Theorem mvlladdcd

Description: Rotate the variables right in an equation with addition on the left, converting it into a subtraction. Version of mvlladdd with a commuted consequent, and of mvrladdd with a commuted hypothesis. (Contributed by SN, 21-Aug-2024)

	Ref	Expression
Hypotheses	mvlraddd.1	⊢ ( 𝜑 → 𝐴 ∈ ℂ )
	mvlraddd.2	⊢ ( 𝜑 → 𝐵 ∈ ℂ )
	mvlraddd.3	⊢ ( 𝜑 → ( 𝐴 + 𝐵 ) = 𝐶 )
Assertion	mvlladdcd	⊢ ( 𝜑 → ( 𝐶 − 𝐴 ) = 𝐵 )

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	1 2 3	mvlladdd	⊢ ( 𝜑 → 𝐵 = ( 𝐶 − 𝐴 ) )
5	4	eqcomd	⊢ ( 𝜑 → ( 𝐶 − 𝐴 ) = 𝐵 )