ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  freq2 Unicode version

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.
StepHypRef Expression
1 frforeq2 4392 . . 3  |-  ( A  =  B  ->  (FrFor  R A s  <-> FrFor  R B s ) )
21albidv 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 )
52, 3, 43bitr4g 223 1  |-  ( A  =  B  ->  ( R  Fr  A  <->  R  Fr  B ) )
Colors of variables: wff set class
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
  Copyright terms: Public domain W3C validator