Theorem weeq2 4454

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 4443	. . 3 |- ( A = B -> ( R Fr A <-> R Fr B ) )
2		raleq 2730	. . . . 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 2742	. . . 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 2742	. . 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 473	. 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 4431	. 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 4431	. 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 223	1 |- ( A = B -> ( R We A <-> R We B ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1397   A.wral 2510   class class class wbr 4088   Fr wfr 4425   We wwe 4427

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-io 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-ext 2213

This theorem depends on definitions:  df-bi 117  df-tru 1400  df-nf 1509  df-sb 1811  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ral 2515  df-in 3206  df-ss 3213  df-frfor 4428  df-frind 4429  df-wetr 4431

This theorem is referenced by:  reg3exmid  4678