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Theorem caovord 5697
Description: Convert an operation ordering law to class notation. (Contributed by NM, 19-Feb-1996.)
Hypotheses
Ref Expression
caovord.1  |-  A  e. 
_V
caovord.2  |-  B  e. 
_V
caovord.3  |-  ( z  e.  S  ->  (
x R y  <->  ( z F x ) R ( z F y ) ) )
Assertion
Ref Expression
caovord  |-  ( 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    x, F, y, z    x, R, y, z    x, S, y, z

Proof of Theorem caovord
StepHypRef Expression
1 oveq1 5544 . . . 4  |-  ( z  =  C  ->  (
z F A )  =  ( C F A ) )
2 oveq1 5544 . . . 4  |-  ( z  =  C  ->  (
z F B )  =  ( C F B ) )
31, 2breq12d 3800 . . 3  |-  ( z  =  C  ->  (
( z F A ) R ( z F B )  <->  ( C F A ) R ( C F B ) ) )
43bibi2d 230 . 2  |-  ( z  =  C  ->  (
( A R B  <-> 
( z F A ) R ( z F B ) )  <-> 
( A R B  <-> 
( C F A ) R ( C F B ) ) ) )
5 caovord.1 . . 3  |-  A  e. 
_V
6 caovord.2 . . 3  |-  B  e. 
_V
7 breq1 3790 . . . . . 6  |-  ( x  =  A  ->  (
x R y  <->  A R
y ) )
8 oveq2 5545 . . . . . . 7  |-  ( x  =  A  ->  (
z F x )  =  ( z F A ) )
98breq1d 3797 . . . . . 6  |-  ( x  =  A  ->  (
( z F x ) R ( z F y )  <->  ( z F A ) R ( z F y ) ) )
107, 9bibi12d 233 . . . . 5  |-  ( x  =  A  ->  (
( x R y  <-> 
( z F x ) R ( z F y ) )  <-> 
( A R y  <-> 
( z F A ) R ( z F y ) ) ) )
11 breq2 3791 . . . . . 6  |-  ( y  =  B  ->  ( A R y  <->  A R B ) )
12 oveq2 5545 . . . . . . 7  |-  ( y  =  B  ->  (
z F y )  =  ( z F B ) )
1312breq2d 3799 . . . . . 6  |-  ( y  =  B  ->  (
( z F A ) R ( z F y )  <->  ( z F A ) R ( z F B ) ) )
1411, 13bibi12d 233 . . . . 5  |-  ( y  =  B  ->  (
( A R y  <-> 
( z F A ) R ( z F y ) )  <-> 
( A R B  <-> 
( z F A ) R ( z F B ) ) ) )
1510, 14sylan9bb 450 . . . 4  |-  ( ( x  =  A  /\  y  =  B )  ->  ( ( x R y  <->  ( z F x ) R ( z F y ) )  <->  ( A R B  <->  ( z F A ) R ( z F B ) ) ) )
1615imbi2d 228 . . 3  |-  ( ( x  =  A  /\  y  =  B )  ->  ( ( z  e.  S  ->  ( x R y  <->  ( z F x ) R ( z F y ) ) )  <->  ( z  e.  S  ->  ( A R B  <->  ( z F A ) R ( z F B ) ) ) ) )
17 caovord.3 . . 3  |-  ( z  e.  S  ->  (
x R y  <->  ( z F x ) R ( z F y ) ) )
185, 6, 16, 17vtocl2 2655 . 2  |-  ( z  e.  S  ->  ( A R B  <->  ( z F A ) R ( z F B ) ) )
194, 18vtoclga 2665 1  |-  ( C  e.  S  ->  ( A R B  <->  ( C F A ) R ( C F B ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 102    <-> wb 103    = wceq 1285    e. wcel 1434   _Vcvv 2602   class class class wbr 3787  (class class class)co 5537
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2064
This theorem depends on definitions:  df-bi 115  df-3an 922  df-tru 1288  df-nf 1391  df-sb 1687  df-clab 2069  df-cleq 2075  df-clel 2078  df-nfc 2209  df-rex 2355  df-v 2604  df-un 2978  df-sn 3406  df-pr 3407  df-op 3409  df-uni 3604  df-br 3788  df-iota 4891  df-fv 4934  df-ov 5540
This theorem is referenced by:  caovord2  5698  caovord3  5699
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