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Theorem frforeq2 4330
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 2665 . . . . 5  |-  ( A  =  B  ->  ( A. y  e.  A  ( y R x  ->  y  e.  T
)  <->  A. y  e.  B  ( y R x  ->  y  e.  T
) ) )
21imbi1d 230 . . . 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 2673 . . 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 3170 . . 3  |-  ( A  =  B  ->  ( A  C_  T  <->  B  C_  T
) )
53, 4imbi12d 233 . 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 4316 . 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 4316 . 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 222 1  |-  ( A  =  B  ->  (FrFor  R A T  <-> FrFor  R B T ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 104    = wceq 1348    e. wcel 2141   A.wral 2448    C_ wss 3121   class class class wbr 3989  FrFor wfrfor 4312
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-ext 2152
This theorem depends on definitions:  df-bi 116  df-tru 1351  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ral 2453  df-in 3127  df-ss 3134  df-frfor 4316
This theorem is referenced by:  freq2  4331
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