Theorem freq2 4393

Description: Equality theorem for the well-founded predicate. (Contributed by NM, 3-Apr-1994.)

Assertion

Ref	Expression
freq2	|- ( A = B -> ( R Fr A <-> R Fr B ) )

Proof of Theorem freq2

Dummy variable s is distinct from all other variables.

Step	Hyp	Ref	Expression
1		frforeq2 4392	. . 3 |- ( A = B -> (FrFor R A s <-> FrFor R B s ) )
2	1	albidv 1847	. 2 |- ( A = B -> ( A. sFrFor R A s <-> A. sFrFor R B s ) )
3		df-frind 4379	. 2 |- ( R Fr A <-> A. sFrFor R A s )
4		df-frind 4379	. 2 |- ( R Fr B <-> A. sFrFor R B s )
5	2, 3, 4	3bitr4g 223	1 |- ( A = B -> ( R Fr A <-> R Fr B ) )

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-io 711  ax-5 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-10 1528  ax-11 1529  ax-i12 1530  ax-bndl 1532  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-i5r 1558  ax-ext 2187

This theorem depends on definitions:  df-bi 117  df-tru 1376  df-nf 1484  df-sb 1786  df-clab 2192  df-cleq 2198  df-clel 2201  df-nfc 2337  df-ral 2489  df-in 3172  df-ss 3179  df-frfor 4378  df-frind 4379

This theorem is referenced by:  weeq2  4404