Theorem freq2 4305

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

Assertion

Ref	Expression
freq2	|- ( A = B -> ( R Fr A <-> R Fr B ) )

Proof of Theorem freq2

Dummy variable s is distinct from all other variables.

Step	Hyp	Ref	Expression
1		frforeq2 4304	. . 3 |- ( A = B -> (FrFor R A s <-> FrFor R B s ) )
2	1	albidv 1804	. 2 |- ( A = B -> ( A. sFrFor R A s <-> A. sFrFor R B s ) )
3		df-frind 4291	. 2 |- ( R Fr A <-> A. sFrFor R A s )
4		df-frind 4291	. 2 |- ( R Fr B <-> A. sFrFor R B s )
5	2, 3, 4	3bitr4g 222	1 |- ( A = B -> ( R Fr A <-> R Fr B ) )

Syntax hints:   -> wi 4   <-> wb 104   A.wal 1333   = wceq 1335  FrFor wfrfor 4286   Fr wfr 4287

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 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2139

This theorem depends on definitions:  df-bi 116  df-tru 1338  df-nf 1441  df-sb 1743  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-ral 2440  df-in 3108  df-ss 3115  df-frfor 4290  df-frind 4291

This theorem is referenced by:  weeq2  4316