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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 ⊢ ( 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 )