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Theorem freq1 4799
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 4403 . . . . . 6  |-  ( R  =  S  ->  (
z R y  <->  z S
y ) )
21notbid 294 . . . . 5  |-  ( R  =  S  ->  ( -.  z R y  <->  -.  z S y ) )
32rexralbidv 2881 . . . 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 316 . . 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 1680 . 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 4788 . 2  |-  ( R  Fr  A  <->  A. x
( ( x  C_  A  /\  x  =/=  (/) )  ->  E. y  e.  x  A. z  e.  x  -.  z R y ) )
7 df-fr 4788 . 2  |-  ( S  Fr  A  <->  A. x
( ( x  C_  A  /\  x  =/=  (/) )  ->  E. y  e.  x  A. z  e.  x  -.  z S y ) )
85, 6, 73bitr4g 288 1  |-  ( R  =  S  ->  ( R  Fr  A  <->  S  Fr  A ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 184    /\ wa 369   A.wal 1368    = wceq 1370    =/= wne 2648   A.wral 2799   E.wrex 2800    C_ wss 3437   (/)c0 3746   class class class wbr 4401    Fr wfr 4785
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-ext 2432
This theorem depends on definitions:  df-bi 185  df-an 371  df-ex 1588  df-cleq 2446  df-clel 2449  df-ral 2804  df-rex 2805  df-br 4402  df-fr 4788
This theorem is referenced by:  weeq1  4817  freq12d  29540
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