Theorem weeq2 4335

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 4324	. . 3 |- ( A = B -> ( R Fr A <-> R Fr B ) )
2		raleq 2661	. . . . 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 2669	. . . 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 2669	. . 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 465	. 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 4312	. 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 4312	. 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 222	1 |- ( A = B -> ( R We A <-> R We B ) )

Syntax hints:   -> wi 4   /\ wa 103   <-> wb 104   = wceq 1343   A.wral 2444   class class class wbr 3982   Fr wfr 4306   We wwe 4308

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-io 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-ext 2147

This theorem depends on definitions:  df-bi 116  df-tru 1346  df-nf 1449  df-sb 1751  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-ral 2449  df-in 3122  df-ss 3129  df-frfor 4309  df-frind 4310  df-wetr 4312

This theorem is referenced by:  reg3exmid  4557