Theorem freq1 4409

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 4408	. . 3 |- ( R = S -> (FrFor R A s <-> FrFor S A s ) )
2	1	albidv 1848	. 2 |- ( R = S -> ( A. sFrFor R A s <-> A. sFrFor S A s ) )
3		df-frind 4397	. 2 |- ( R Fr A <-> A. sFrFor R A s )
4		df-frind 4397	. 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 4392   Fr wfr 4393

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 1471  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-4 1534  ax-17 1550  ax-ial 1558  ax-ext 2189

This theorem depends on definitions:  df-bi 117  df-nf 1485  df-cleq 2200  df-clel 2203  df-ral 2491  df-br 4060  df-frfor 4396  df-frind 4397

This theorem is referenced by:  weeq1  4421