Metamath Proof Explorer

Theorem cnveq

Description: Equality theorem for converse relation. (Contributed by NM, 13-Aug-1995)

Ref Expression
Assertion cnveq 
|- ( A = B -> `' A = `' B )

Proof

Step Hyp Ref Expression
1 cnvss 
 |-  ( A C_ B -> `' A C_ `' B )
2 cnvss 
 |-  ( B C_ A -> `' B C_ `' A )
3 1 2 anim12i 
 |-  ( ( A C_ B /\ B C_ A ) -> ( `' A C_ `' B /\ `' B C_ `' A ) )
4 eqss 
 |-  ( A = B <-> ( A C_ B /\ B C_ A ) )
5 eqss 
 |-  ( `' A = `' B <-> ( `' A C_ `' B /\ `' B C_ `' A ) )
6 3 4 5 3imtr4i 
 |-  ( A = B -> `' A = `' B )