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Metamath Proof Explorer

Theorem mvrladdd

Description: Move the left 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 mvrladdd ( 𝜑 → ( 𝐴 − 𝐵 ) = 𝐶 )

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 1 2 3 comraddd ⊢ ( 𝜑 → 𝐴 = ( 𝐶 + 𝐵 ) )
5 2 1 4 mvrraddd ⊢ ( 𝜑 → ( 𝐴 − 𝐵 ) = 𝐶 )