Theorem cmcmii

Description: Commutation is symmetric. Theorem 2(v) of Kalmbach p. 22. (Contributed by NM, 7-Aug-2004) (New usage is discouraged.)

	Ref	Expression
Hypotheses	pjoml2.1	⊢ 𝐴 ∈ Cℋ
	pjoml2.2	⊢ 𝐵 ∈ Cℋ
	cmcmi.1	⊢ 𝐴 𝐶ℋ 𝐵
Assertion	cmcmii	⊢ 𝐵 𝐶ℋ 𝐴

Proof

Step	Hyp	Ref	Expression
1		pjoml2.1	⊢ 𝐴 ∈ Cℋ
2		pjoml2.2	⊢ 𝐵 ∈ Cℋ
3		cmcmi.1	⊢ 𝐴 𝐶ℋ 𝐵
4	1 2	cmcmi	⊢ ( 𝐴 𝐶ℋ 𝐵 ↔ 𝐵 𝐶ℋ 𝐴 )
5	3 4	mpbi	⊢ 𝐵 𝐶ℋ 𝐴