Theorem freq2 4346

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 4345	. . 3 |- ( A = B -> (FrFor R A s <-> FrFor R B s ) )
2	1	albidv 1824	. 2 |- ( A = B -> ( A. sFrFor R A s <-> A. sFrFor R B s ) )
3		df-frind 4332	. 2 |- ( R Fr A <-> A. sFrFor R A s )
4		df-frind 4332	. 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 1351   = wceq 1353  FrFor wfrfor 4327   Fr wfr 4328

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 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-ext 2159

This theorem depends on definitions:  df-bi 117  df-tru 1356  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-in 3135  df-ss 3142  df-frfor 4331  df-frind 4332

This theorem is referenced by:  weeq2  4357