Theorem freq1 4391

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 4390	. . 3 |- ( R = S -> (FrFor R A s <-> FrFor S A s ) )
2	1	albidv 1847	. 2 |- ( R = S -> ( A. sFrFor R A s <-> A. sFrFor S A s ) )
3		df-frind 4379	. 2 |- ( R Fr A <-> A. sFrFor R A s )
4		df-frind 4379	. 2 |- ( S Fr A <-> A. sFrFor S A s )
5	2, 3, 4	3bitr4g 223	1 |- ( R = S -> ( R Fr A <-> S Fr A ) )

Syntax hints:   -> wi 4   <-> wb 105   A.wal 1371   = wceq 1373  FrFor wfrfor 4374   Fr wfr 4375

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-5 1470  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-4 1533  ax-17 1549  ax-ial 1557  ax-ext 2187

This theorem depends on definitions:  df-bi 117  df-nf 1484  df-cleq 2198  df-clel 2201  df-ral 2489  df-br 4045  df-frfor 4378  df-frind 4379

This theorem is referenced by:  weeq1  4403