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Metamath Proof Explorer

Theorem mvlladdd

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

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

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 2 1 pncand ⊢ ( 𝜑 → ( ( 𝐵 + 𝐴 ) − 𝐴 ) = 𝐵 )
5 1 2 addcomd ⊢ ( 𝜑 → ( 𝐴 + 𝐵 ) = ( 𝐵 + 𝐴 ) )
6 5 3 eqtr3d ⊢ ( 𝜑 → ( 𝐵 + 𝐴 ) = 𝐶 )
7 6 oveq1d ⊢ ( 𝜑 → ( ( 𝐵 + 𝐴 ) − 𝐴 ) = ( 𝐶 − 𝐴 ) )
8 4 7 eqtr3d ⊢ ( 𝜑 → 𝐵 = ( 𝐶 − 𝐴 ) )