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Theorem frforeq2 4363
Description: Equality theorem for the well-founded predicate. (Contributed by Jim Kingdon, 22-Sep-2021.)
Assertion
Ref Expression
frforeq2  |-  ( A  =  B  ->  (FrFor  R A T  <-> FrFor  R B T ) )

Proof of Theorem frforeq2
Dummy variables  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 raleq 2686 . . . . 5  |-  ( A  =  B  ->  ( A. y  e.  A  ( y R x  ->  y  e.  T
)  <->  A. y  e.  B  ( y R x  ->  y  e.  T
) ) )
21imbi1d 231 . . . 4  |-  ( A  =  B  ->  (
( A. y  e.  A  ( y R x  ->  y  e.  T )  ->  x  e.  T )  <->  ( A. y  e.  B  (
y R x  -> 
y  e.  T )  ->  x  e.  T
) ) )
32raleqbi1dv 2694 . . 3  |-  ( A  =  B  ->  ( A. x  e.  A  ( A. y  e.  A  ( y R x  ->  y  e.  T
)  ->  x  e.  T )  <->  A. x  e.  B  ( A. y  e.  B  (
y R x  -> 
y  e.  T )  ->  x  e.  T
) ) )
4 sseq1 3193 . . 3  |-  ( A  =  B  ->  ( A  C_  T  <->  B  C_  T
) )
53, 4imbi12d 234 . 2  |-  ( A  =  B  ->  (
( A. x  e.  A  ( A. y  e.  A  ( y R x  ->  y  e.  T )  ->  x  e.  T )  ->  A  C_  T )  <->  ( A. x  e.  B  ( A. y  e.  B  ( y R x  ->  y  e.  T
)  ->  x  e.  T )  ->  B  C_  T ) ) )
6 df-frfor 4349 . 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
) )
7 df-frfor 4349 . 2  |-  (FrFor  R B T  <->  ( A. x  e.  B  ( A. y  e.  B  (
y R x  -> 
y  e.  T )  ->  x  e.  T
)  ->  B  C_  T
) )
85, 6, 73bitr4g 223 1  |-  ( A  =  B  ->  (FrFor  R A T  <-> FrFor  R B T ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 105    = wceq 1364    e. wcel 2160   A.wral 2468    C_ wss 3144   class class class wbr 4018  FrFor wfrfor 4345
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 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2171
This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ral 2473  df-in 3150  df-ss 3157  df-frfor 4349
This theorem is referenced by:  freq2  4364
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