Theorem weeq1 4341

Description: Equality theorem for the well-ordering predicate. (Contributed by NM, 9-Mar-1997.)

Assertion

Ref	Expression
weeq1	|- ( R = S -> ( R We A <-> S We A ) )

Proof of Theorem weeq1

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

Step	Hyp	Ref	Expression
1		freq1 4329	. . 3 |- ( R = S -> ( R Fr A <-> S Fr A ) )
2		breq 3991	. . . . . . . 8 |- ( R = S -> ( x R y <-> x S y ) )
3		breq 3991	. . . . . . . 8 |- ( R = S -> ( y R z <-> y S z ) )
4	2, 3	anbi12d 470	. . . . . . 7 |- ( R = S -> ( ( x R y /\ y R z ) <-> ( x S y /\ y S z ) ) )
5		breq 3991	. . . . . . 7 |- ( R = S -> ( x R z <-> x S z ) )
6	4, 5	imbi12d 233	. . . . . 6 |- ( R = S -> ( ( ( x R y /\ y R z ) -> x R z ) <-> ( ( x S y /\ y S z ) -> x S z ) ) )
7	6	ralbidv 2470	. . . . 5 |- ( R = S -> ( A. z e. A ( ( x R y /\ y R z ) -> x R z ) <-> A. z e. A ( ( x S y /\ y S z ) -> x S z ) ) )
8	7	ralbidv 2470	. . . 4 |- ( R = S -> ( A. y e. A A. z e. A ( ( x R y /\ y R z ) -> x R z ) <-> A. y e. A A. z e. A ( ( x S y /\ y S z ) -> x S z ) ) )
9	8	ralbidv 2470	. . 3 |- ( R = S -> ( A. x e. A A. y e. A A. z e. A ( ( x R y /\ y R z ) -> x R z ) <-> A. x e. A A. y e. A A. z e. A ( ( x S y /\ y S z ) -> x S z ) ) )
10	1, 9	anbi12d 470	. 2 |- ( R = S -> ( ( R Fr A /\ A. x e. A A. y e. A A. z e. A ( ( x R y /\ y R z ) -> x R z ) ) <-> ( S Fr A /\ A. x e. A A. y e. A A. z e. A ( ( x S y /\ y S z ) -> x S z ) ) ) )
11		df-wetr 4319	. 2 |- ( R We A <-> ( R Fr A /\ A. x e. A A. y e. A A. z e. A ( ( x R y /\ y R z ) -> x R z ) ) )
12		df-wetr 4319	. 2 |- ( S We A <-> ( S Fr A /\ A. x e. A A. y e. A A. z e. A ( ( x S y /\ y S z ) -> x S z ) ) )
13	10, 11, 12	3bitr4g 222	1 |- ( R = S -> ( R We A <-> S We A ) )

Syntax hints:   -> wi 4   /\ wa 103   <-> wb 104   = wceq 1348   A.wral 2448   class class class wbr 3989   Fr wfr 4313   We wwe 4315

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-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-frfor 4316  df-frind 4317  df-wetr 4319

This theorem is referenced by: (None)