Theorem freq2 4381

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 4380	. . 3 |- ( A = B -> (FrFor R A s <-> FrFor R B s ) )
2	1	albidv 1838	. 2 |- ( A = B -> ( A. sFrFor R A s <-> A. sFrFor R B s ) )
3		df-frind 4367	. 2 |- ( R Fr A <-> A. sFrFor R A s )
4		df-frind 4367	. 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 1362   = wceq 1364  FrFor wfrfor 4362   Fr wfr 4363

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 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-ext 2178

This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1475  df-sb 1777  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ral 2480  df-in 3163  df-ss 3170  df-frfor 4366  df-frind 4367

This theorem is referenced by:  weeq2  4392