Theorem weeq2 4279

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 4268	. . 3 |- ( A = B -> ( R Fr A <-> R Fr B ) )
2		raleq 2626	. . . . 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 2634	. . . 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 2634	. . 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 464	. 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 4256	. 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 4256	. 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 1331   A.wral 2416   class class class wbr 3929   Fr wfr 4250   We wwe 4252

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 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121

This theorem depends on definitions:  df-bi 116  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ral 2421  df-in 3077  df-ss 3084  df-frfor 4253  df-frind 4254  df-wetr 4256

This theorem is referenced by:  reg3exmid  4494