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Theorem caovord 6499
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 6315 . . . 4  |-  ( z  =  C  ->  (
z F A )  =  ( C F A ) )
2 oveq1 6315 . . . 4  |-  ( z  =  C  ->  (
z F B )  =  ( C F B ) )
31, 2breq12d 4408 . . 3  |-  ( z  =  C  ->  (
( z F A ) R ( z F B )  <->  ( C F A ) R ( C F B ) ) )
43bibi2d 325 . 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 4398 . . . . . 6  |-  ( x  =  A  ->  (
x R y  <->  A R
y ) )
8 oveq2 6316 . . . . . . 7  |-  ( x  =  A  ->  (
z F x )  =  ( z F A ) )
98breq1d 4405 . . . . . 6  |-  ( x  =  A  ->  (
( z F x ) R ( z F y )  <->  ( z F A ) R ( z F y ) ) )
107, 9bibi12d 328 . . . . 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 4399 . . . . . 6  |-  ( y  =  B  ->  ( A R y  <->  A R B ) )
12 oveq2 6316 . . . . . . 7  |-  ( y  =  B  ->  (
z F y )  =  ( z F B ) )
1312breq2d 4407 . . . . . 6  |-  ( y  =  B  ->  (
( z F A ) R ( z F y )  <->  ( z F A ) R ( z F B ) ) )
1411, 13bibi12d 328 . . . . 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 714 . . . 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 323 . . 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 3088 . 2  |-  ( z  e.  S  ->  ( A R B  <->  ( z F A ) R ( z F B ) ) )
194, 18vtoclga 3099 1  |-  ( C  e.  S  ->  ( A R B  <->  ( C F A ) R ( C F B ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 189    /\ wa 376    = wceq 1452    e. wcel 1904   _Vcvv 3031   class class class wbr 4395  (class class class)co 6308
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451
This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-3an 1009  df-tru 1455  df-ex 1672  df-nf 1676  df-sb 1806  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-rex 2762  df-rab 2765  df-v 3033  df-dif 3393  df-un 3395  df-in 3397  df-ss 3404  df-nul 3723  df-if 3873  df-sn 3960  df-pr 3962  df-op 3966  df-uni 4191  df-br 4396  df-iota 5553  df-fv 5597  df-ov 6311
This theorem is referenced by:  caovord2  6500  caovord3  6501  genpcl  9451
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