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Metamath Proof Explorer

Theorem comraddd

Description: Commute RHS addition, in deduction form. (Contributed by David A. Wheeler, 11-Oct-2018)

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

Proof

Step Hyp Ref Expression
1 comraddd.1 ⊢ ( 𝜑 → 𝐵 ∈ ℂ )
2 comraddd.2 ⊢ ( 𝜑 → 𝐶 ∈ ℂ )
3 comraddd.3 ⊢ ( 𝜑 → 𝐴 = ( 𝐵 + 𝐶 ) )
4 1 2 addcomd ⊢ ( 𝜑 → ( 𝐵 + 𝐶 ) = ( 𝐶 + 𝐵 ) )
5 3 4 eqtrd ⊢ ( 𝜑 → 𝐴 = ( 𝐶 + 𝐵 ) )