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Theorem caovord 6471
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 6288 . . . 4  |-  ( z  =  C  ->  (
z F A )  =  ( C F A ) )
2 oveq1 6288 . . . 4  |-  ( z  =  C  ->  (
z F B )  =  ( C F B ) )
31, 2breq12d 4450 . . 3  |-  ( z  =  C  ->  (
( z F A ) R ( z F B )  <->  ( C F A ) R ( C F B ) ) )
43bibi2d 318 . 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 4440 . . . . . 6  |-  ( x  =  A  ->  (
x R y  <->  A R
y ) )
8 oveq2 6289 . . . . . . 7  |-  ( x  =  A  ->  (
z F x )  =  ( z F A ) )
98breq1d 4447 . . . . . 6  |-  ( x  =  A  ->  (
( z F x ) R ( z F y )  <->  ( z F A ) R ( z F y ) ) )
107, 9bibi12d 321 . . . . 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 4441 . . . . . 6  |-  ( y  =  B  ->  ( A R y  <->  A R B ) )
12 oveq2 6289 . . . . . . 7  |-  ( y  =  B  ->  (
z F y )  =  ( z F B ) )
1312breq2d 4449 . . . . . 6  |-  ( y  =  B  ->  (
( z F A ) R ( z F y )  <->  ( z F A ) R ( z F B ) ) )
1411, 13bibi12d 321 . . . . 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 699 . . . 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 316 . . 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 3148 . 2  |-  ( z  e.  S  ->  ( A R B  <->  ( z F A ) R ( z F B ) ) )
194, 18vtoclga 3159 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 184    /\ wa 369    = wceq 1383    e. wcel 1804   _Vcvv 3095   class class class wbr 4437  (class class class)co 6281
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1605  ax-4 1618  ax-5 1691  ax-6 1734  ax-7 1776  ax-10 1823  ax-11 1828  ax-12 1840  ax-13 1985  ax-ext 2421
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 976  df-tru 1386  df-ex 1600  df-nf 1604  df-sb 1727  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2593  df-rex 2799  df-rab 2802  df-v 3097  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3771  df-if 3927  df-sn 4015  df-pr 4017  df-op 4021  df-uni 4235  df-br 4438  df-iota 5541  df-fv 5586  df-ov 6284
This theorem is referenced by:  caovord2  6472  caovord3  6473  genpcl  9389
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