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

Proof

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