Theorem poeq1 4054

Description: Equality theorem for partial ordering predicate. (Contributed by NM, 27-Mar-1997.)

Assertion

Ref	Expression
poeq1	|- ( R = S -> ( R Po A <-> S Po A ) )

Proof of Theorem poeq1

Dummy variables x y z are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		breq 3787	. . . . . 6 |- ( R = S -> ( x R x <-> x S x ) )
2	1	notbid 624	. . . . 5 |- ( R = S -> ( -. x R x <-> -. x S x ) )
3		breq 3787	. . . . . . 7 |- ( R = S -> ( x R y <-> x S y ) )
4		breq 3787	. . . . . . 7 |- ( R = S -> ( y R z <-> y S z ) )
5	3, 4	anbi12d 456	. . . . . 6 |- ( R = S -> ( ( x R y /\ y R z ) <-> ( x S y /\ y S z ) ) )
6		breq 3787	. . . . . 6 |- ( R = S -> ( x R z <-> x S z ) )
7	5, 6	imbi12d 232	. . . . 5 |- ( R = S -> ( ( ( x R y /\ y R z ) -> x R z ) <-> ( ( x S y /\ y S z ) -> x S z ) ) )
8	2, 7	anbi12d 456	. . . 4 |- ( R = S -> ( ( -. x R x /\ ( ( x R y /\ y R z ) -> x R z ) ) <-> ( -. x S x /\ ( ( x S y /\ y S z ) -> x S z ) ) ) )
9	8	ralbidv 2368	. . 3 |- ( R = S -> ( A. z e. A ( -. x R x /\ ( ( x R y /\ y R z ) -> x R z ) ) <-> A. z e. A ( -. x S x /\ ( ( x S y /\ y S z ) -> x S z ) ) ) )
10	9	2ralbidv 2390	. 2 |- ( R = S -> ( A. x e. A A. y e. A A. z e. A ( -. x R x /\ ( ( x R y /\ y R z ) -> x R z ) ) <-> A. x e. A A. y e. A A. z e. A ( -. x S x /\ ( ( x S y /\ y S z ) -> x S z ) ) ) )
11		df-po 4051	. 2 |- ( R Po A <-> A. x e. A A. y e. A A. z e. A ( -. x R x /\ ( ( x R y /\ y R z ) -> x R z ) ) )
12		df-po 4051	. 2 |- ( S Po A <-> A. x e. A A. y e. A A. z e. A ( -. x S x /\ ( ( x S y /\ y S z ) -> x S z ) ) )
13	10, 11, 12	3bitr4g 221	1 |- ( R = S -> ( R Po A <-> S Po A ) )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 102   <-> wb 103   = wceq 1284   A.wral 2348   class class class wbr 3785   Po wpo 4049

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 576  ax-in2 577  ax-5 1376  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-4 1440  ax-17 1459  ax-ial 1467  ax-ext 2063

This theorem depends on definitions:  df-bi 115  df-nf 1390  df-cleq 2074  df-clel 2077  df-ral 2353  df-br 3786  df-po 4051

This theorem is referenced by:  soeq1  4070