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Theorem caovordig 6040
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 2560 . 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 4007 . . . 4  |-  ( x  =  A  ->  (
x R y  <->  A R
y ) )
4 oveq2 5883 . . . . 5  |-  ( x  =  A  ->  (
z F x )  =  ( z F A ) )
54breq1d 4014 . . . 4  |-  ( x  =  A  ->  (
( z F x ) R ( z F y )  <->  ( z F A ) R ( z F y ) ) )
63, 5imbi12d 234 . . 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 4008 . . . 4  |-  ( y  =  B  ->  ( A R y  <->  A R B ) )
8 oveq2 5883 . . . . 5  |-  ( y  =  B  ->  (
z F y )  =  ( z F B ) )
98breq2d 4016 . . . 4  |-  ( y  =  B  ->  (
( z F A ) R ( z F y )  <->  ( z F A ) R ( z F B ) ) )
107, 9imbi12d 234 . . 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 5882 . . . . 5  |-  ( z  =  C  ->  (
z F A )  =  ( C F A ) )
12 oveq1 5882 . . . . 5  |-  ( z  =  C  ->  (
z F B )  =  ( C F B ) )
1311, 12breq12d 4017 . . . 4  |-  ( z  =  C  ->  (
( z F A ) R ( z F B )  <->  ( C F A ) R ( C F B ) ) )
1413imbi2d 230 . . 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 2858 . 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 281 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 104    /\ w3a 978    = wceq 1353    e. wcel 2148   A.wral 2455   class class class wbr 4004  (class class class)co 5875
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-ext 2159
This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-rex 2461  df-v 2740  df-un 3134  df-sn 3599  df-pr 3600  df-op 3602  df-uni 3811  df-br 4005  df-iota 5179  df-fv 5225  df-ov 5878
This theorem is referenced by:  caovordid  6041
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