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Theorem caovordg 6213
Description: Convert an operation ordering law to class notation. (Contributed by NM, 19-Feb-1996.) (Revised by Mario Carneiro, 30-Dec-2014.)
Hypothesis
Ref Expression
caovordg.1  |-  ( (
ph  /\  ( x  e.  S  /\  y  e.  S  /\  z  e.  S ) )  -> 
( x R y  <-> 
( z F x ) R ( z F y ) ) )
Assertion
Ref Expression
caovordg  |-  ( (
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 caovordg
StepHypRef Expression
1 caovordg.1 . . 3  |-  ( (
ph  /\  ( x  e.  S  /\  y  e.  S  /\  z  e.  S ) )  -> 
( x R y  <-> 
( z F x ) R ( z F y ) ) )
21ralrimivvva 2759 . 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 4175 . . . 4  |-  ( x  =  A  ->  (
x R y  <->  A R
y ) )
4 oveq2 6048 . . . . 5  |-  ( x  =  A  ->  (
z F x )  =  ( z F A ) )
54breq1d 4182 . . . 4  |-  ( x  =  A  ->  (
( z F x ) R ( z F y )  <->  ( z F A ) R ( z F y ) ) )
63, 5bibi12d 313 . . 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 4176 . . . 4  |-  ( y  =  B  ->  ( A R y  <->  A R B ) )
8 oveq2 6048 . . . . 5  |-  ( y  =  B  ->  (
z F y )  =  ( z F B ) )
98breq2d 4184 . . . 4  |-  ( y  =  B  ->  (
( z F A ) R ( z F y )  <->  ( z F A ) R ( z F B ) ) )
107, 9bibi12d 313 . . 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 6047 . . . . 5  |-  ( z  =  C  ->  (
z F A )  =  ( C F A ) )
12 oveq1 6047 . . . . 5  |-  ( z  =  C  ->  (
z F B )  =  ( C F B ) )
1311, 12breq12d 4185 . . . 4  |-  ( z  =  C  ->  (
( z F A ) R ( z F B )  <->  ( C F A ) R ( C F B ) ) )
1413bibi2d 310 . . 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 3021 . 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 456 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    <-> wb 177    /\ wa 359    /\ w3a 936    = wceq 1649    e. wcel 1721   A.wral 2666   class class class wbr 4172  (class class class)co 6040
This theorem is referenced by:  caovordd  6214
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ral 2671  df-rex 2672  df-rab 2675  df-v 2918  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-nul 3589  df-if 3700  df-sn 3780  df-pr 3781  df-op 3783  df-uni 3976  df-br 4173  df-iota 5377  df-fv 5421  df-ov 6043
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