Theorem poeq1 4344

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 4045	. . . . . 6 |- ( R = S -> ( x R x <-> x S x ) )
2	1	notbid 668	. . . . 5 |- ( R = S -> ( -. x R x <-> -. x S x ) )
3		breq 4045	. . . . . . 7 |- ( R = S -> ( x R y <-> x S y ) )
4		breq 4045	. . . . . . 7 |- ( R = S -> ( y R z <-> y S z ) )
5	3, 4	anbi12d 473	. . . . . 6 |- ( R = S -> ( ( x R y /\ y R z ) <-> ( x S y /\ y S z ) ) )
6		breq 4045	. . . . . 6 |- ( R = S -> ( x R z <-> x S z ) )
7	5, 6	imbi12d 234	. . . . 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 473	. . . 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 2505	. . 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 2529	. 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 4341	. 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 4341	. 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 223	1 |- ( R = S -> ( R Po A <-> S Po A ) )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1372   A.wral 2483   class class class wbr 4043   Po wpo 4339

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-5 1469  ax-gen 1471  ax-ie1 1515  ax-ie2 1516  ax-4 1532  ax-17 1548  ax-ial 1556  ax-ext 2186

This theorem depends on definitions:  df-bi 117  df-nf 1483  df-cleq 2197  df-clel 2200  df-ral 2488  df-br 4044  df-po 4341

This theorem is referenced by:  soeq1  4360