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Metamath Proof Explorer

Theorem mvlraddd

Description: Move the right 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 mvlraddd ( 𝜑 → 𝐴 = ( 𝐶 − 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 pncand ⊢ ( 𝜑 → ( ( 𝐴 + 𝐵 ) − 𝐵 ) = 𝐴 )
5 3 oveq1d ⊢ ( 𝜑 → ( ( 𝐴 + 𝐵 ) − 𝐵 ) = ( 𝐶 − 𝐵 ) )
6 4 5 eqtr3d ⊢ ( 𝜑 → 𝐴 = ( 𝐶 − 𝐵 ) )