Theorem freq1 4322

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 4321	. . 3 |- ( R = S -> (FrFor R A s <-> FrFor S A s ) )
2	1	albidv 1812	. 2 |- ( R = S -> ( A. sFrFor R A s <-> A. sFrFor S A s ) )
3		df-frind 4310	. 2 |- ( R Fr A <-> A. sFrFor R A s )
4		df-frind 4310	. 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 1341   = wceq 1343  FrFor wfrfor 4305   Fr wfr 4306

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 1435  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-4 1498  ax-17 1514  ax-ial 1522  ax-ext 2147

This theorem depends on definitions:  df-bi 116  df-nf 1449  df-cleq 2158  df-clel 2161  df-ral 2449  df-br 3983  df-frfor 4309  df-frind 4310

This theorem is referenced by:  weeq1  4334