Theorem freq1 4390

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 4389	. . 3 |- ( R = S -> (FrFor R A s <-> FrFor S A s ) )
2	1	albidv 1846	. 2 |- ( R = S -> ( A. sFrFor R A s <-> A. sFrFor S A s ) )
3		df-frind 4378	. 2 |- ( R Fr A <-> A. sFrFor R A s )
4		df-frind 4378	. 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 1370   = wceq 1372  FrFor wfrfor 4373   Fr wfr 4374

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 1469  ax-gen 1471  ax-ie1 1515  ax-ie2 1516  ax-4 1532  ax-17 1548  ax-ial 1556  ax-ext 2186

This theorem depends on definitions:  df-bi 117  df-nf 1483  df-cleq 2197  df-clel 2200  df-ral 2488  df-br 4044  df-frfor 4377  df-frind 4378

This theorem is referenced by:  weeq1  4402