MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  freq1 Unicode version

Theorem freq1 4363
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 4025 . . . . . 6  |-  ( R  =  S  ->  (
z R y  <->  z S
y ) )
21notbid 285 . . . . 5  |-  ( R  =  S  ->  ( -.  z R y  <->  -.  z S y ) )
32rexralbidv 2587 . . . 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 307 . . 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 1611 . 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 4352 . 2  |-  ( R  Fr  A  <->  A. x
( ( x  C_  A  /\  x  =/=  (/) )  ->  E. y  e.  x  A. z  e.  x  -.  z R y ) )
7 df-fr 4352 . 2  |-  ( S  Fr  A  <->  A. x
( ( x  C_  A  /\  x  =/=  (/) )  ->  E. y  e.  x  A. z  e.  x  -.  z S y ) )
85, 6, 73bitr4g 279 1  |-  ( R  =  S  ->  ( R  Fr  A  <->  S  Fr  A ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 176    /\ wa 358   A.wal 1527    = wceq 1623    =/= wne 2446   A.wral 2543   E.wrex 2544    C_ wss 3152   (/)c0 3455   class class class wbr 4023    Fr wfr 4349
This theorem is referenced by:  weeq1  4381  freq12d  27135
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-11 1715  ax-ext 2264
This theorem depends on definitions:  df-bi 177  df-an 360  df-ex 1529  df-nf 1532  df-cleq 2276  df-clel 2279  df-ral 2548  df-rex 2549  df-br 4024  df-fr 4352
  Copyright terms: Public domain W3C validator