Theorem add32i

Description: Commutative/associative law that swaps the last two terms in a triple sum. (Contributed by NM, 21-Jan-1997)

	Ref	Expression
Hypotheses	add.1	⊢ 𝐴 ∈ ℂ
	add.2	⊢ 𝐵 ∈ ℂ
	add.3	⊢ 𝐶 ∈ ℂ
Assertion	add32i	⊢ ( ( 𝐴 + 𝐵 ) + 𝐶 ) = ( ( 𝐴 + 𝐶 ) + 𝐵 )

Proof

Step	Hyp	Ref	Expression
1		add.1	⊢ 𝐴 ∈ ℂ
2		add.2	⊢ 𝐵 ∈ ℂ
3		add.3	⊢ 𝐶 ∈ ℂ
4		add32	⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ ) → ( ( 𝐴 + 𝐵 ) + 𝐶 ) = ( ( 𝐴 + 𝐶 ) + 𝐵 ) )
5	1 2 3 4	mp3an	⊢ ( ( 𝐴 + 𝐵 ) + 𝐶 ) = ( ( 𝐴 + 𝐶 ) + 𝐵 )