Theorem weeq2 4112

Description: Equality theorem for the well-ordering predicate. (Contributed by NM, 3-Apr-1994.)

Assertion

Ref	Expression
weeq2	|- ( A = B -> ( R We A <-> R We B ) )

Proof of Theorem weeq2

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

Step	Hyp	Ref	Expression
1		freq2 4101	. . 3 |- ( A = B -> ( R Fr A <-> R Fr B ) )
2		raleq 2549	. . . . 5 |- ( A = B -> ( A. z e. A ( ( x R y /\ y R z ) -> x R z ) <-> A. z e. B ( ( x R y /\ y R z ) -> x R z ) ) )
3	2	raleqbi1dv 2557	. . . 4 |- ( A = B -> ( A. y e. A A. z e. A ( ( x R y /\ y R z ) -> x R z ) <-> A. y e. B A. z e. B ( ( x R y /\ y R z ) -> x R z ) ) )
4	3	raleqbi1dv 2557	. . 3 |- ( A = B -> ( A. x e. A A. y e. A A. z e. A ( ( x R y /\ y R z ) -> x R z ) <-> A. x e. B A. y e. B A. z e. B ( ( x R y /\ y R z ) -> x R z ) ) )
5	1, 4	anbi12d 456	. 2 |- ( A = B -> ( ( R Fr A /\ A. x e. A A. y e. A A. z e. A ( ( x R y /\ y R z ) -> x R z ) ) <-> ( R Fr B /\ A. x e. B A. y e. B A. z e. B ( ( x R y /\ y R z ) -> x R z ) ) ) )
6		df-wetr 4089	. 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 ) ) )
7		df-wetr 4089	. 2 |- ( R We B <-> ( R Fr B /\ A. x e. B A. y e. B A. z e. B ( ( x R y /\ y R z ) -> x R z ) ) )
8	5, 6, 7	3bitr4g 221	1 |- ( A = B -> ( R We A <-> R We B ) )

Syntax hints:   -> wi 4   /\ wa 102   <-> wb 103   = wceq 1284   A.wral 2348   class class class wbr 3785   Fr wfr 4083   We wwe 4085

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-bndl 1439  ax-4 1440  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063

This theorem depends on definitions:  df-bi 115  df-tru 1287  df-nf 1390  df-sb 1686  df-clab 2068  df-cleq 2074  df-clel 2077  df-nfc 2208  df-ral 2353  df-in 2979  df-ss 2986  df-frfor 4086  df-frind 4087  df-wetr 4089

This theorem is referenced by:  reg3exmid  4322