ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  frforeq3 Unicode version

Theorem frforeq3 4320
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 2228 . . . . . . 7  |-  ( S  =  T  ->  (
y  e.  S  <->  y  e.  T ) )
21imbi2d 229 . . . . . 6  |-  ( S  =  T  ->  (
( y R x  ->  y  e.  S
)  <->  ( y R x  ->  y  e.  T ) ) )
32ralbidv 2464 . . . . 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 2228 . . . . 5  |-  ( S  =  T  ->  (
x  e.  S  <->  x  e.  T ) )
53, 4imbi12d 233 . . . 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 2464 . . 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 3162 . . 3  |-  ( S  =  T  ->  ( A  C_  S  <->  A  C_  T
) )
86, 7imbi12d 233 . 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 4304 . 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 4304 . 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 222 1  |-  ( S  =  T  ->  (FrFor  R A S  <-> FrFor  R A T ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 104    = wceq 1342    e. wcel 2135   A.wral 2442    C_ wss 3112   class class class wbr 3977  FrFor wfrfor 4300
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-5 1434  ax-7 1435  ax-gen 1436  ax-ie1 1480  ax-ie2 1481  ax-8 1491  ax-11 1493  ax-4 1497  ax-17 1513  ax-i9 1517  ax-ial 1521  ax-i5r 1522  ax-ext 2146
This theorem depends on definitions:  df-bi 116  df-nf 1448  df-sb 1750  df-clab 2151  df-cleq 2157  df-clel 2160  df-ral 2447  df-in 3118  df-ss 3125  df-frfor 4304
This theorem is referenced by:  frind  4325
  Copyright terms: Public domain W3C validator