Theorem freq1 4329

Description: Equality theorem for the well-founded predicate. (Contributed by NM, 9-Mar-1997.)

Assertion

Ref	Expression
freq1	|- ( R = S -> ( R Fr A <-> S Fr A ) )

Proof of Theorem freq1

Dummy variable s is distinct from all other variables.

Step	Hyp	Ref	Expression
1		frforeq1 4328	. . 3 |- ( R = S -> (FrFor R A s <-> FrFor S A s ) )
2	1	albidv 1817	. 2 |- ( R = S -> ( A. sFrFor R A s <-> A. sFrFor S A s ) )
3		df-frind 4317	. 2 |- ( R Fr A <-> A. sFrFor R A s )
4		df-frind 4317	. 2 |- ( S Fr A <-> A. sFrFor S A s )
5	2, 3, 4	3bitr4g 222	1 |- ( R = S -> ( R Fr A <-> S Fr A ) )

Syntax hints:   -> wi 4   <-> wb 104   A.wal 1346   = wceq 1348  FrFor wfrfor 4312   Fr wfr 4313

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-5 1440  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-4 1503  ax-17 1519  ax-ial 1527  ax-ext 2152

This theorem depends on definitions:  df-bi 116  df-nf 1454  df-cleq 2163  df-clel 2166  df-ral 2453  df-br 3990  df-frfor 4316  df-frind 4317

This theorem is referenced by:  weeq1  4341