Metamath Proof Explorer

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 e. CH
pjoml2.2 
|- B e. CH
Assertion cmcmi 
|- ( A C_H B <-> B C_H A )

Proof

Step Hyp Ref Expression
1 pjoml2.1 
 |-  A e. CH
2 pjoml2.2 
 |-  B e. CH
3 1 2 cmcmlem 
 |-  ( A C_H B -> B C_H A )
4 2 1 cmcmlem 
 |-  ( B C_H A -> A C_H B )
5 3 4 impbii 
 |-  ( A C_H B <-> B C_H A )