This is an inofficial mirror of http://metamath.tirix.org for personal testing of a visualizer extension only.

Metamath Proof Explorer

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 ) ) )