Theorem weeq2 4351

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 4340	. . 3 |- ( A = B -> ( R Fr A <-> R Fr B ) )
2		raleq 2670	. . . . 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 2678	. . . 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 2678	. . 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 4328	. 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 4328	. 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 1353   A.wral 2453   class class class wbr 3998   Fr wfr 4322   We wwe 4324

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 709  ax-5 1445  ax-7 1446  ax-gen 1447  ax-ie1 1491  ax-ie2 1492  ax-8 1502  ax-10 1503  ax-11 1504  ax-i12 1505  ax-bndl 1507  ax-4 1508  ax-17 1524  ax-i9 1528  ax-ial 1532  ax-i5r 1533  ax-ext 2157

This theorem depends on definitions:  df-bi 117  df-tru 1356  df-nf 1459  df-sb 1761  df-clab 2162  df-cleq 2168  df-clel 2171  df-nfc 2306  df-ral 2458  df-in 3133  df-ss 3140  df-frfor 4325  df-frind 4326  df-wetr 4328

This theorem is referenced by:  reg3exmid  4573