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Theorem caovordig 5943
Description: Convert an operation ordering law to class notation. (Contributed by Mario Carneiro, 31-Dec-2014.)
Hypothesis
Ref Expression
caovordig.1  |-  ( (
ph  /\  ( x  e.  S  /\  y  e.  S  /\  z  e.  S ) )  -> 
( x R y  ->  ( z F x ) R ( z F y ) ) )
Assertion
Ref Expression
caovordig  |-  ( (
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 caovordig
StepHypRef Expression
1 caovordig.1 . . 3  |-  ( (
ph  /\  ( x  e.  S  /\  y  e.  S  /\  z  e.  S ) )  -> 
( x R y  ->  ( z F x ) R ( z F y ) ) )
21ralrimivvva 2518 . 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 3939 . . . 4  |-  ( x  =  A  ->  (
x R y  <->  A R
y ) )
4 oveq2 5789 . . . . 5  |-  ( x  =  A  ->  (
z F x )  =  ( z F A ) )
54breq1d 3946 . . . 4  |-  ( x  =  A  ->  (
( z F x ) R ( z F y )  <->  ( z F A ) R ( z F y ) ) )
63, 5imbi12d 233 . . 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 3940 . . . 4  |-  ( y  =  B  ->  ( A R y  <->  A R B ) )
8 oveq2 5789 . . . . 5  |-  ( y  =  B  ->  (
z F y )  =  ( z F B ) )
98breq2d 3948 . . . 4  |-  ( y  =  B  ->  (
( z F A ) R ( z F y )  <->  ( z F A ) R ( z F B ) ) )
107, 9imbi12d 233 . . 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 5788 . . . . 5  |-  ( z  =  C  ->  (
z F A )  =  ( C F A ) )
12 oveq1 5788 . . . . 5  |-  ( z  =  C  ->  (
z F B )  =  ( C F B ) )
1311, 12breq12d 3949 . . . 4  |-  ( z  =  C  ->  (
( z F A ) R ( z F B )  <->  ( C F A ) R ( C F B ) ) )
1413imbi2d 229 . . 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 2808 . 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 279 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    /\ wa 103    /\ w3a 963    = wceq 1332    e. wcel 1481   A.wral 2417   class class class wbr 3936  (class class class)co 5781
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122
This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1335  df-nf 1438  df-sb 1737  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-ral 2422  df-rex 2423  df-v 2691  df-un 3079  df-sn 3537  df-pr 3538  df-op 3540  df-uni 3744  df-br 3937  df-iota 5095  df-fv 5138  df-ov 5784
This theorem is referenced by:  caovordid  5944
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