Theorem freq2 4263

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 4262	. . 3 |- ( A = B -> (FrFor R A s <-> FrFor R B s ) )
2	1	albidv 1796	. 2 |- ( A = B -> ( A. sFrFor R A s <-> A. sFrFor R B s ) )
3		df-frind 4249	. 2 |- ( R Fr A <-> A. sFrFor R A s )
4		df-frind 4249	. 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 1329   = wceq 1331  FrFor wfrfor 4244   Fr wfr 4245

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 2119

This theorem depends on definitions:  df-bi 116  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2124  df-cleq 2130  df-clel 2133  df-nfc 2268  df-ral 2419  df-in 3072  df-ss 3079  df-frfor 4248  df-frind 4249

This theorem is referenced by:  weeq2  4274