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Theorem freq1 4486
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 4148 . . . . . 6  |-  ( R  =  S  ->  (
z R y  <->  z S
y ) )
21notbid 286 . . . . 5  |-  ( R  =  S  ->  ( -.  z R y  <->  -.  z S y ) )
32rexralbidv 2686 . . . 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 1632 . 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 4475 . 2  |-  ( R  Fr  A  <->  A. x
( ( x  C_  A  /\  x  =/=  (/) )  ->  E. y  e.  x  A. z  e.  x  -.  z R y ) )
7 df-fr 4475 . 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 1546    = wceq 1649    =/= wne 2543   A.wral 2642   E.wrex 2643    C_ wss 3256   (/)c0 3564   class class class wbr 4146    Fr wfr 4472
This theorem is referenced by:  weeq1  4504  freq12d  26797
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-11 1753  ax-ext 2361
This theorem depends on definitions:  df-bi 178  df-an 361  df-ex 1548  df-nf 1551  df-cleq 2373  df-clel 2376  df-ral 2647  df-rex 2648  df-br 4147  df-fr 4475
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