Metamath Proof Explorer

Theorem cgrcom

Description: Congruence commutes between the two sides. (Contributed by Scott Fenton, 12-Jun-2013)

Ref Expression
Assertion cgrcom 
|- ( ( N e. NN /\ ( A e. ( EE ` N ) /\ B e. ( EE ` N ) ) /\ ( C e. ( EE ` N ) /\ D e. ( EE ` N ) ) ) -> ( <. A , B >. Cgr <. C , D >. <-> <. C , D >. Cgr <. A , B >. ) )

Proof

Step Hyp Ref Expression
1 cgrcomim 
 |-  ( ( N e. NN /\ ( A e. ( EE ` N ) /\ B e. ( EE ` N ) ) /\ ( C e. ( EE ` N ) /\ D e. ( EE ` N ) ) ) -> ( <. A , B >. Cgr <. C , D >. -> <. C , D >. Cgr <. A , B >. ) )
2 cgrcomim 
 |-  ( ( N e. NN /\ ( C e. ( EE ` N ) /\ D e. ( EE ` N ) ) /\ ( A e. ( EE ` N ) /\ B e. ( EE ` N ) ) ) -> ( <. C , D >. Cgr <. A , B >. -> <. A , B >. Cgr <. C , D >. ) )
3 2 3com23 
 |-  ( ( N e. NN /\ ( A e. ( EE ` N ) /\ B e. ( EE ` N ) ) /\ ( C e. ( EE ` N ) /\ D e. ( EE ` N ) ) ) -> ( <. C , D >. Cgr <. A , B >. -> <. A , B >. Cgr <. C , D >. ) )
4 1 3 impbid 
 |-  ( ( N e. NN /\ ( A e. ( EE ` N ) /\ B e. ( EE ` N ) ) /\ ( C e. ( EE ` N ) /\ D e. ( EE ` N ) ) ) -> ( <. A , B >. Cgr <. C , D >. <-> <. C , D >. Cgr <. A , B >. ) )