Metamath Proof Explorer

Theorem soeq2

Description: Equality theorem for the strict ordering predicate. (Contributed by NM, 16-Mar-1997)

Ref Expression
Assertion soeq2 
|- ( A = B -> ( R Or A <-> R Or B ) )

Proof

Step Hyp Ref Expression
1 soss 
 |-  ( A C_ B -> ( R Or B -> R Or A ) )
2 soss 
 |-  ( B C_ A -> ( R Or A -> R Or B ) )
3 1 2 anim12i 
 |-  ( ( A C_ B /\ B C_ A ) -> ( ( R Or B -> R Or A ) /\ ( R Or A -> R Or B ) ) )
4 eqss 
 |-  ( A = B <-> ( A C_ B /\ B C_ A ) )
5 dfbi2 
 |-  ( ( R Or B <-> R Or A ) <-> ( ( R Or B -> R Or A ) /\ ( R Or A -> R Or B ) ) )
6 3 4 5 3imtr4i 
 |-  ( A = B -> ( R Or B <-> R Or A ) )
7 6 bicomd 
 |-  ( A = B -> ( R Or A <-> R Or B ) )