Theorem cmcmi

Description: Commutation is symmetric. Theorem 2(v) of Kalmbach p. 22. (Contributed by NM, 4-Nov-2000) (New usage is discouraged.)

	Ref	Expression
Hypotheses	pjoml2.1	⊢ 𝐴 ∈ Cℋ
	pjoml2.2	⊢ 𝐵 ∈ Cℋ
Assertion	cmcmi	⊢ ( 𝐴 𝐶ℋ 𝐵 ↔ 𝐵 𝐶ℋ 𝐴 )

Proof

Step	Hyp	Ref	Expression
1		pjoml2.1	⊢ 𝐴 ∈ Cℋ
2		pjoml2.2	⊢ 𝐵 ∈ Cℋ
3	1 2	cmcmlem	⊢ ( 𝐴 𝐶ℋ 𝐵 → 𝐵 𝐶ℋ 𝐴 )
4	2 1	cmcmlem	⊢ ( 𝐵 𝐶ℋ 𝐴 → 𝐴 𝐶ℋ 𝐵 )
5	3 4	impbii	⊢ ( 𝐴 𝐶ℋ 𝐵 ↔ 𝐵 𝐶ℋ 𝐴 )