Theorem soeq1

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

	Ref	Expression
Assertion	soeq1	|- ( R = S -> ( R Or A <-> S Or A ) )

Proof

Step	Hyp	Ref	Expression
1		poeq1	|- ( R = S -> ( R Po A <-> S Po A ) )
2		breq	|- ( R = S -> ( x R y <-> x S y ) )
3		biidd	|- ( R = S -> ( x = y <-> x = y ) )
4		breq	|- ( R = S -> ( y R x <-> y S x ) )
5	2 3 4	3orbi123d	|- ( R = S -> ( ( x R y \/ x = y \/ y R x ) <-> ( x S y \/ x = y \/ y S x ) ) )
6	5	2ralbidv	|- ( R = S -> ( A. x e. A A. y e. A ( x R y \/ x = y \/ y R x ) <-> A. x e. A A. y e. A ( x S y \/ x = y \/ y S x ) ) )
7	1 6	anbi12d	|- ( R = S -> ( ( R Po A /\ A. x e. A A. y e. A ( x R y \/ x = y \/ y R x ) ) <-> ( S Po A /\ A. x e. A A. y e. A ( x S y \/ x = y \/ y S x ) ) ) )
8		df-so	|- ( R Or A <-> ( R Po A /\ A. x e. A A. y e. A ( x R y \/ x = y \/ y R x ) ) )
9		df-so	|- ( S Or A <-> ( S Po A /\ A. x e. A A. y e. A ( x S y \/ x = y \/ y S x ) ) )
10	7 8 9	3bitr4g	|- ( R = S -> ( R Or A <-> S Or A ) )