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

Theorem caovordg 6027
Description: Convert an operation ordering law to class notation. (Contributed by NM, 19-Feb-1996.) (Revised by Mario Carneiro, 30-Dec-2014.)
Hypothesis
Ref Expression
caovordg.1  |-  ( (
ph  /\  ( x  e.  S  /\  y  e.  S  /\  z  e.  S ) )  -> 
( x R y  <-> 
( z F x ) R ( z F y ) ) )
Assertion
Ref Expression
caovordg  |-  ( (
ph  /\  ( A  e.  S  /\  B  e.  S  /\  C  e.  S ) )  -> 
( A R B  <-> 
( C F A ) R ( C F B ) ) )
Distinct variable groups:    x, y,
z, A    x, B, y, z    x, C, y, z    ph, x, y, z   
x, F, y, z   
x, R, y, z   
x, S, y, z

Proof of Theorem caovordg
StepHypRef Expression
1 caovordg.1 . . 3  |-  ( (
ph  /\  ( x  e.  S  /\  y  e.  S  /\  z  e.  S ) )  -> 
( x R y  <-> 
( z F x ) R ( z F y ) ) )
21ralrimivvva 2636 . 2  |-  ( ph  ->  A. x  e.  S  A. y  e.  S  A. z  e.  S  ( x R y  <-> 
( z F x ) R ( z F y ) ) )
3 breq1 4026 . . . 4  |-  ( x  =  A  ->  (
x R y  <->  A R
y ) )
4 oveq2 5866 . . . . 5  |-  ( x  =  A  ->  (
z F x )  =  ( z F A ) )
54breq1d 4033 . . . 4  |-  ( x  =  A  ->  (
( z F x ) R ( z F y )  <->  ( z F A ) R ( z F y ) ) )
63, 5bibi12d 312 . . 3  |-  ( x  =  A  ->  (
( x R y  <-> 
( z F x ) R ( z F y ) )  <-> 
( A R y  <-> 
( z F A ) R ( z F y ) ) ) )
7 breq2 4027 . . . 4  |-  ( y  =  B  ->  ( A R y  <->  A R B ) )
8 oveq2 5866 . . . . 5  |-  ( y  =  B  ->  (
z F y )  =  ( z F B ) )
98breq2d 4035 . . . 4  |-  ( y  =  B  ->  (
( z F A ) R ( z F y )  <->  ( z F A ) R ( z F B ) ) )
107, 9bibi12d 312 . . 3  |-  ( y  =  B  ->  (
( A R y  <-> 
( z F A ) R ( z F y ) )  <-> 
( A R B  <-> 
( z F A ) R ( z F B ) ) ) )
11 oveq1 5865 . . . . 5  |-  ( z  =  C  ->  (
z F A )  =  ( C F A ) )
12 oveq1 5865 . . . . 5  |-  ( z  =  C  ->  (
z F B )  =  ( C F B ) )
1311, 12breq12d 4036 . . . 4  |-  ( z  =  C  ->  (
( z F A ) R ( z F B )  <->  ( C F A ) R ( C F B ) ) )
1413bibi2d 309 . . 3  |-  ( z  =  C  ->  (
( A R B  <-> 
( z F A ) R ( z F B ) )  <-> 
( A R B  <-> 
( C F A ) R ( C F B ) ) ) )
156, 10, 14rspc3v 2893 . 2  |-  ( ( A  e.  S  /\  B  e.  S  /\  C  e.  S )  ->  ( A. x  e.  S  A. y  e.  S  A. z  e.  S  ( x R y  <->  ( z F x ) R ( z F y ) )  ->  ( A R B  <->  ( C F A ) R ( C F B ) ) ) )
162, 15mpan9 455 1  |-  ( (
ph  /\  ( A  e.  S  /\  B  e.  S  /\  C  e.  S ) )  -> 
( A R B  <-> 
( C F A ) R ( C F B ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    /\ w3a 934    = wceq 1623    e. wcel 1684   A.wral 2543   class class class wbr 4023  (class class class)co 5858
This theorem is referenced by:  caovordd  6028
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-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-br 4024  df-iota 5219  df-fv 5263  df-ov 5861
  Copyright terms: Public domain W3C validator