Theorem soeq1 4232

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 4216	. . 3 |- ( R = S -> ( R Po A <-> S Po A ) )
2		breq 3926	. . . . . 6 |- ( R = S -> ( x R y <-> x S y ) )
3		breq 3926	. . . . . . 7 |- ( R = S -> ( x R z <-> x S z ) )
4		breq 3926	. . . . . . 7 |- ( R = S -> ( z R y <-> z S y ) )
5	3, 4	orbi12d 782	. . . . . 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 2457	. . . 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 2435	. . 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 464	. 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 4214	. 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 4214	. 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 697   = wceq 1331   A.wral 2414   class class class wbr 3924   Po wpo 4211   Or wor 4212

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 603  ax-in2 604  ax-io 698  ax-5 1423  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-4 1487  ax-17 1506  ax-ial 1514  ax-ext 2119

This theorem depends on definitions:  df-bi 116  df-nf 1437  df-cleq 2130  df-clel 2133  df-ral 2419  df-br 3925  df-po 4213  df-iso 4214

This theorem is referenced by: (None)