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Theorem caovordd 6256
Description: Convert an operation ordering law to class notation. (Contributed by Mario Carneiro, 30-Dec-2014.)
Hypotheses
Ref Expression
caovordg.1  |-  ( (
ph  /\  ( x  e.  S  /\  y  e.  S  /\  z  e.  S ) )  -> 
( x R y  <-> 
( z F x ) R ( z F y ) ) )
caovordd.2  |-  ( ph  ->  A  e.  S )
caovordd.3  |-  ( ph  ->  B  e.  S )
caovordd.4  |-  ( ph  ->  C  e.  S )
Assertion
Ref Expression
caovordd  |-  ( ph  ->  ( 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 caovordd
StepHypRef Expression
1 id 21 . 2  |-  ( ph  ->  ph )
2 caovordd.2 . 2  |-  ( ph  ->  A  e.  S )
3 caovordd.3 . 2  |-  ( ph  ->  B  e.  S )
4 caovordd.4 . 2  |-  ( ph  ->  C  e.  S )
5 caovordg.1 . . 3  |-  ( (
ph  /\  ( x  e.  S  /\  y  e.  S  /\  z  e.  S ) )  -> 
( x R y  <-> 
( z F x ) R ( z F y ) ) )
65caovordg 6255 . 2  |-  ( (
ph  /\  ( A  e.  S  /\  B  e.  S  /\  C  e.  S ) )  -> 
( A R B  <-> 
( C F A ) R ( C F B ) ) )
71, 2, 3, 4, 6syl13anc 1187 1  |-  ( ph  ->  ( A R B  <-> 
( C F A ) R ( C F B ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 178    /\ wa 360    /\ w3a 937    e. wcel 1726   class class class wbr 4213  (class class class)co 6082
This theorem is referenced by:  caovord2d  6257  caovord3d  6258
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2418
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-clab 2424  df-cleq 2430  df-clel 2433  df-nfc 2562  df-ral 2711  df-rex 2712  df-rab 2715  df-v 2959  df-dif 3324  df-un 3326  df-in 3328  df-ss 3335  df-nul 3630  df-if 3741  df-sn 3821  df-pr 3822  df-op 3824  df-uni 4017  df-br 4214  df-iota 5419  df-fv 5463  df-ov 6085
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