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Theorem freq1 4544
Description: Equality theorem for the well-founded predicate. (Contributed by NM, 9-Mar-1997.)
Assertion
Ref Expression
freq1  |-  ( R  =  S  ->  ( R  Fr  A  <->  S  Fr  A ) )

Proof of Theorem freq1
Dummy variables  x  y  z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 breq 4206 . . . . . 6  |-  ( R  =  S  ->  (
z R y  <->  z S
y ) )
21notbid 286 . . . . 5  |-  ( R  =  S  ->  ( -.  z R y  <->  -.  z S y ) )
32rexralbidv 2741 . . . 4  |-  ( R  =  S  ->  ( E. y  e.  x  A. z  e.  x  -.  z R y  <->  E. y  e.  x  A. z  e.  x  -.  z S y ) )
43imbi2d 308 . . 3  |-  ( R  =  S  ->  (
( ( x  C_  A  /\  x  =/=  (/) )  ->  E. y  e.  x  A. z  e.  x  -.  z R y )  <-> 
( ( x  C_  A  /\  x  =/=  (/) )  ->  E. y  e.  x  A. z  e.  x  -.  z S y ) ) )
54albidv 1635 . 2  |-  ( R  =  S  ->  ( A. x ( ( x 
C_  A  /\  x  =/=  (/) )  ->  E. y  e.  x  A. z  e.  x  -.  z R y )  <->  A. x
( ( x  C_  A  /\  x  =/=  (/) )  ->  E. y  e.  x  A. z  e.  x  -.  z S y ) ) )
6 df-fr 4533 . 2  |-  ( R  Fr  A  <->  A. x
( ( x  C_  A  /\  x  =/=  (/) )  ->  E. y  e.  x  A. z  e.  x  -.  z R y ) )
7 df-fr 4533 . 2  |-  ( S  Fr  A  <->  A. x
( ( x  C_  A  /\  x  =/=  (/) )  ->  E. y  e.  x  A. z  e.  x  -.  z S y ) )
85, 6, 73bitr4g 280 1  |-  ( R  =  S  ->  ( R  Fr  A  <->  S  Fr  A ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 177    /\ wa 359   A.wal 1549    = wceq 1652    =/= wne 2598   A.wral 2697   E.wrex 2698    C_ wss 3312   (/)c0 3620   class class class wbr 4204    Fr wfr 4530
This theorem is referenced by:  weeq1  4562  freq12d  27067
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-11 1761  ax-ext 2416
This theorem depends on definitions:  df-bi 178  df-an 361  df-ex 1551  df-nf 1554  df-cleq 2428  df-clel 2431  df-ral 2702  df-rex 2703  df-br 4205  df-fr 4533
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