Theorem soeq1 4350

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 4334	. . 3 |- ( R = S -> ( R Po A <-> S Po A ) )
2		breq 4035	. . . . . 6 |- ( R = S -> ( x R y <-> x S y ) )
3		breq 4035	. . . . . . 7 |- ( R = S -> ( x R z <-> x S z ) )
4		breq 4035	. . . . . . 7 |- ( R = S -> ( z R y <-> z S y ) )
5	3, 4	orbi12d 794	. . . . . 6 |- ( R = S -> ( ( x R z \/ z R y ) <-> ( x S z \/ z S y ) ) )
6	2, 5	imbi12d 234	. . . . 5 |- ( R = S -> ( ( x R y -> ( x R z \/ z R y ) ) <-> ( x S y -> ( x S z \/ z S y ) ) ) )
7	6	2ralbidv 2521	. . . 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 2497	. . 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 473	. 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 4332	. 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 4332	. 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 223	1 |- ( R = S -> ( R Or A <-> S Or A ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   \/ wo 709   = wceq 1364   A.wral 2475   class class class wbr 4033   Po wpo 4329   Or wor 4330

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-io 710  ax-5 1461  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-4 1524  ax-17 1540  ax-ial 1548  ax-ext 2178

This theorem depends on definitions:  df-bi 117  df-nf 1475  df-cleq 2189  df-clel 2192  df-ral 2480  df-br 4034  df-po 4331  df-iso 4332

This theorem is referenced by: (None)