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Theorem caovordid 6253
Description: Convert an operation ordering law to class notation. (Contributed by Mario Carneiro, 31-Dec-2014.)
Hypotheses
Ref Expression
caovordig.1  |-  ( (
ph  /\  ( x  e.  S  /\  y  e.  S  /\  z  e.  S ) )  -> 
( x R y  ->  ( z F x ) R ( z F y ) ) )
caovordid.2  |-  ( ph  ->  A  e.  S )
caovordid.3  |-  ( ph  ->  B  e.  S )
caovordid.4  |-  ( ph  ->  C  e.  S )
Assertion
Ref Expression
caovordid  |-  ( 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 caovordid
StepHypRef Expression
1 id 20 . 2  |-  ( ph  ->  ph )
2 caovordid.2 . 2  |-  ( ph  ->  A  e.  S )
3 caovordid.3 . 2  |-  ( ph  ->  B  e.  S )
4 caovordid.4 . 2  |-  ( ph  ->  C  e.  S )
5 caovordig.1 . . 3  |-  ( (
ph  /\  ( x  e.  S  /\  y  e.  S  /\  z  e.  S ) )  -> 
( x R y  ->  ( z F x ) R ( z F y ) ) )
65caovordig 6252 . 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 1186 1  |-  ( ph  ->  ( A R B  ->  ( C F A ) R ( C F B ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    /\ w3a 936    e. wcel 1725   class class class wbr 4212  (class class class)co 6081
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ral 2710  df-rex 2711  df-rab 2714  df-v 2958  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-nul 3629  df-if 3740  df-sn 3820  df-pr 3821  df-op 3823  df-uni 4016  df-br 4213  df-iota 5418  df-fv 5462  df-ov 6084
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