Theorem poeq1 4036

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 3766	. . . . . 6 |- ( R = S -> ( x R x <-> x S x ) )
2	1	notbid 592	. . . . 5 |- ( R = S -> ( -. x R x <-> -. x S x ) )
3		breq 3766	. . . . . . 7 |- ( R = S -> ( x R y <-> x S y ) )
4		breq 3766	. . . . . . 7 |- ( R = S -> ( y R z <-> y S z ) )
5	3, 4	anbi12d 442	. . . . . 6 |- ( R = S -> ( ( x R y /\ y R z ) <-> ( x S y /\ y S z ) ) )
6		breq 3766	. . . . . 6 |- ( R = S -> ( x R z <-> x S z ) )
7	5, 6	imbi12d 223	. . . . 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 442	. . . 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 2326	. . 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 2348	. 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 4033	. 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 4033	. 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 212	1 |- ( R = S -> ( R Po A <-> S Po A ) )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 97   <-> wb 98   = wceq 1243   A.wral 2306   class class class wbr 3764   Po wpo 4031

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-5 1336  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-4 1400  ax-17 1419  ax-ial 1427  ax-ext 2022

This theorem depends on definitions:  df-bi 110  df-nf 1350  df-cleq 2033  df-clel 2036  df-ral 2311  df-br 3765  df-po 4033

This theorem is referenced by:  soeq1  4052