Theorem add12d

Description: Commutative/associative law that swaps the first two terms in a triple sum. (Contributed by Mario Carneiro, 27-May-2016)

	Ref	Expression
Hypotheses	addd.1	⊢ ( 𝜑 → 𝐴 ∈ ℂ )
	addd.2	⊢ ( 𝜑 → 𝐵 ∈ ℂ )
	addd.3	⊢ ( 𝜑 → 𝐶 ∈ ℂ )
Assertion	add12d	⊢ ( 𝜑 → ( 𝐴 + ( 𝐵 + 𝐶 ) ) = ( 𝐵 + ( 𝐴 + 𝐶 ) ) )

Proof

Step	Hyp	Ref	Expression
1		addd.1	⊢ ( 𝜑 → 𝐴 ∈ ℂ )
2		addd.2	⊢ ( 𝜑 → 𝐵 ∈ ℂ )
3		addd.3	⊢ ( 𝜑 → 𝐶 ∈ ℂ )
4		add12	⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ ) → ( 𝐴 + ( 𝐵 + 𝐶 ) ) = ( 𝐵 + ( 𝐴 + 𝐶 ) ) )
5	1 2 3 4	syl3anc	⊢ ( 𝜑 → ( 𝐴 + ( 𝐵 + 𝐶 ) ) = ( 𝐵 + ( 𝐴 + 𝐶 ) ) )