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Theorem caovordig 6041
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 2649 . 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 4042 . . . 4  |-  ( x  =  A  ->  (
x R y  <->  A R
y ) )
4 oveq2 5882 . . . . 5  |-  ( x  =  A  ->  (
z F x )  =  ( z F A ) )
54breq1d 4049 . . . 4  |-  ( x  =  A  ->  (
( z F x ) R ( z F y )  <->  ( z F A ) R ( z F y ) ) )
63, 5imbi12d 311 . . 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 4043 . . . 4  |-  ( y  =  B  ->  ( A R y  <->  A R B ) )
8 oveq2 5882 . . . . 5  |-  ( y  =  B  ->  (
z F y )  =  ( z F B ) )
98breq2d 4051 . . . 4  |-  ( y  =  B  ->  (
( z F A ) R ( z F y )  <->  ( z F A ) R ( z F B ) ) )
107, 9imbi12d 311 . . 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 5881 . . . . 5  |-  ( z  =  C  ->  (
z F A )  =  ( C F A ) )
12 oveq1 5881 . . . . 5  |-  ( z  =  C  ->  (
z F B )  =  ( C F B ) )
1311, 12breq12d 4052 . . . 4  |-  ( z  =  C  ->  (
( z F A ) R ( z F B )  <->  ( C F A ) R ( C F B ) ) )
1413imbi2d 307 . . 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 2906 . 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 455 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 358    /\ w3a 934    = wceq 1632    e. wcel 1696   A.wral 2556   class class class wbr 4039  (class class class)co 5874
This theorem is referenced by:  caovordid  6042
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ral 2561  df-rex 2562  df-rab 2565  df-v 2803  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-br 4040  df-iota 5235  df-fv 5279  df-ov 5877
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