Theorem soeq1 4300

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

Assertion

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

Proof of Theorem soeq1

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

Step	Hyp	Ref	Expression
1		poeq1 4284	. . 3 |- ( R = S -> ( R Po A <-> S Po A ) )
2		breq 3991	. . . . . 6 |- ( R = S -> ( x R y <-> x S y ) )
3		breq 3991	. . . . . . 7 |- ( R = S -> ( x R z <-> x S z ) )
4		breq 3991	. . . . . . 7 |- ( R = S -> ( z R y <-> z S y ) )
5	3, 4	orbi12d 788	. . . . . 6 |- ( R = S -> ( ( x R z \/ z R y ) <-> ( x S z \/ z S y ) ) )
6	2, 5	imbi12d 233	. . . . 5 |- ( R = S -> ( ( x R y -> ( x R z \/ z R y ) ) <-> ( x S y -> ( x S z \/ z S y ) ) ) )
7	6	2ralbidv 2494	. . . 4 |- ( R = S -> ( A. y e. A A. z e. A ( x R y -> ( x R z \/ z R y ) ) <-> A. y e. A A. z e. A ( x S y -> ( x S z \/ z S y ) ) ) )
8	7	ralbidv 2470	. . 3 |- ( R = S -> ( A. x e. A A. y e. A A. z e. A ( x R y -> ( x R z \/ z R y ) ) <-> A. x e. A A. y e. A A. z e. A ( x S y -> ( x S z \/ z S y ) ) ) )
9	1, 8	anbi12d 470	. 2 |- ( R = S -> ( ( R Po A /\ A. x e. A A. y e. A A. z e. A ( x R y -> ( x R z \/ z R y ) ) ) <-> ( S Po A /\ A. x e. A A. y e. A A. z e. A ( x S y -> ( x S z \/ z S y ) ) ) ) )
10		df-iso 4282	. 2 |- ( R Or A <-> ( R Po A /\ A. x e. A A. y e. A A. z e. A ( x R y -> ( x R z \/ z R y ) ) ) )
11		df-iso 4282	. 2 |- ( S Or A <-> ( S Po A /\ A. x e. A A. y e. A A. z e. A ( x S y -> ( x S z \/ z S y ) ) ) )
12	9, 10, 11	3bitr4g 222	1 |- ( R = S -> ( R Or A <-> S Or A ) )

Syntax hints:   -> wi 4   /\ wa 103   <-> wb 104   \/ wo 703   = wceq 1348   A.wral 2448   class class class wbr 3989   Po wpo 4279   Or wor 4280

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 609  ax-in2 610  ax-io 704  ax-5 1440  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-4 1503  ax-17 1519  ax-ial 1527  ax-ext 2152

This theorem depends on definitions:  df-bi 116  df-nf 1454  df-cleq 2163  df-clel 2166  df-ral 2453  df-br 3990  df-po 4281  df-iso 4282

This theorem is referenced by: (None)