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Theorem frforeq3 4110
Description: Equality theorem for the well-founded predicate. (Contributed by Jim Kingdon, 22-Sep-2021.)
Assertion
Ref Expression
frforeq3  |-  ( S  =  T  ->  (FrFor  R A S  <-> FrFor  R A T ) )

Proof of Theorem frforeq3
Dummy variables  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eleq2 2143 . . . . . . 7  |-  ( S  =  T  ->  (
y  e.  S  <->  y  e.  T ) )
21imbi2d 228 . . . . . 6  |-  ( S  =  T  ->  (
( y R x  ->  y  e.  S
)  <->  ( y R x  ->  y  e.  T ) ) )
32ralbidv 2369 . . . . 5  |-  ( S  =  T  ->  ( A. y  e.  A  ( y R x  ->  y  e.  S
)  <->  A. y  e.  A  ( y R x  ->  y  e.  T
) ) )
4 eleq2 2143 . . . . 5  |-  ( S  =  T  ->  (
x  e.  S  <->  x  e.  T ) )
53, 4imbi12d 232 . . . 4  |-  ( S  =  T  ->  (
( A. y  e.  A  ( y R x  ->  y  e.  S )  ->  x  e.  S )  <->  ( A. y  e.  A  (
y R x  -> 
y  e.  T )  ->  x  e.  T
) ) )
65ralbidv 2369 . . 3  |-  ( S  =  T  ->  ( A. x  e.  A  ( A. y  e.  A  ( y R x  ->  y  e.  S
)  ->  x  e.  S )  <->  A. x  e.  A  ( A. y  e.  A  (
y R x  -> 
y  e.  T )  ->  x  e.  T
) ) )
7 sseq2 3022 . . 3  |-  ( S  =  T  ->  ( A  C_  S  <->  A  C_  T
) )
86, 7imbi12d 232 . 2  |-  ( S  =  T  ->  (
( A. x  e.  A  ( A. y  e.  A  ( y R x  ->  y  e.  S )  ->  x  e.  S )  ->  A  C_  S )  <->  ( A. x  e.  A  ( A. y  e.  A  ( y R x  ->  y  e.  T
)  ->  x  e.  T )  ->  A  C_  T ) ) )
9 df-frfor 4094 . 2  |-  (FrFor  R A S  <->  ( A. x  e.  A  ( A. y  e.  A  (
y R x  -> 
y  e.  S )  ->  x  e.  S
)  ->  A  C_  S
) )
10 df-frfor 4094 . 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
) )
118, 9, 103bitr4g 221 1  |-  ( S  =  T  ->  (FrFor  R A S  <-> FrFor  R A T ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 103    = wceq 1285    e. wcel 1434   A.wral 2349    C_ wss 2974   class class class wbr 3793  FrFor wfrfor 4090
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 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-11 1438  ax-4 1441  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2064
This theorem depends on definitions:  df-bi 115  df-nf 1391  df-sb 1687  df-clab 2069  df-cleq 2075  df-clel 2078  df-ral 2354  df-in 2980  df-ss 2987  df-frfor 4094
This theorem is referenced by:  frind  4115
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