Theorem mul12d

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

	Ref	Expression
Hypotheses	muld.1	⊢ ( 𝜑 → 𝐴 ∈ ℂ )
	addcomd.2	⊢ ( 𝜑 → 𝐵 ∈ ℂ )
	addcand.3	⊢ ( 𝜑 → 𝐶 ∈ ℂ )
Assertion	mul12d	⊢ ( 𝜑 → ( 𝐴 · ( 𝐵 · 𝐶 ) ) = ( 𝐵 · ( 𝐴 · 𝐶 ) ) )

Proof

Step	Hyp	Ref	Expression
1		muld.1	⊢ ( 𝜑 → 𝐴 ∈ ℂ )
2		addcomd.2	⊢ ( 𝜑 → 𝐵 ∈ ℂ )
3		addcand.3	⊢ ( 𝜑 → 𝐶 ∈ ℂ )
4		mul12	⊢ ( ( 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 ) ) )