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Theorem frforeq1 4168
Description: Equality theorem for the well-founded predicate. (Contributed by Jim Kingdon, 22-Sep-2021.)
Assertion
Ref Expression
frforeq1  |-  ( R  =  S  ->  (FrFor  R A T  <-> FrFor  S A T ) )

Proof of Theorem frforeq1
Dummy variables  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 breq 3845 . . . . . . 7  |-  ( R  =  S  ->  (
y R x  <->  y S x ) )
21imbi1d 229 . . . . . 6  |-  ( R  =  S  ->  (
( y R x  ->  y  e.  T
)  <->  ( y S x  ->  y  e.  T ) ) )
32ralbidv 2380 . . . . 5  |-  ( R  =  S  ->  ( A. y  e.  A  ( y R x  ->  y  e.  T
)  <->  A. y  e.  A  ( y S x  ->  y  e.  T
) ) )
43imbi1d 229 . . . 4  |-  ( R  =  S  ->  (
( A. y  e.  A  ( y R x  ->  y  e.  T )  ->  x  e.  T )  <->  ( A. y  e.  A  (
y S x  -> 
y  e.  T )  ->  x  e.  T
) ) )
54ralbidv 2380 . . 3  |-  ( R  =  S  ->  ( A. x  e.  A  ( A. y  e.  A  ( y R x  ->  y  e.  T
)  ->  x  e.  T )  <->  A. x  e.  A  ( A. y  e.  A  (
y S x  -> 
y  e.  T )  ->  x  e.  T
) ) )
65imbi1d 229 . 2  |-  ( R  =  S  ->  (
( A. x  e.  A  ( A. y  e.  A  ( y R x  ->  y  e.  T )  ->  x  e.  T )  ->  A  C_  T )  <->  ( A. x  e.  A  ( A. y  e.  A  ( y S x  ->  y  e.  T
)  ->  x  e.  T )  ->  A  C_  T ) ) )
7 df-frfor 4156 . 2  |-  (FrFor  R A T  <->  ( A. x  e.  A  ( A. y  e.  A  (
y R x  -> 
y  e.  T )  ->  x  e.  T
)  ->  A  C_  T
) )
8 df-frfor 4156 . 2  |-  (FrFor  S A T  <->  ( A. x  e.  A  ( A. y  e.  A  (
y S x  -> 
y  e.  T )  ->  x  e.  T
)  ->  A  C_  T
) )
96, 7, 83bitr4g 221 1  |-  ( R  =  S  ->  (FrFor  R A T  <-> FrFor  S A T ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 103    = wceq 1289    e. wcel 1438   A.wral 2359    C_ wss 2999   class class class wbr 3843  FrFor wfrfor 4152
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-5 1381  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-4 1445  ax-17 1464  ax-ial 1472  ax-ext 2070
This theorem depends on definitions:  df-bi 115  df-nf 1395  df-cleq 2081  df-clel 2084  df-ral 2364  df-br 3844  df-frfor 4156
This theorem is referenced by:  freq1  4169
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