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Theorem caovordig 6280
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 2821 . 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 4307 . . . 4  |-  ( x  =  A  ->  (
x R y  <->  A R
y ) )
4 oveq2 6111 . . . . 5  |-  ( x  =  A  ->  (
z F x )  =  ( z F A ) )
54breq1d 4314 . . . 4  |-  ( x  =  A  ->  (
( z F x ) R ( z F y )  <->  ( z F A ) R ( z F y ) ) )
63, 5imbi12d 320 . . 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 4308 . . . 4  |-  ( y  =  B  ->  ( A R y  <->  A R B ) )
8 oveq2 6111 . . . . 5  |-  ( y  =  B  ->  (
z F y )  =  ( z F B ) )
98breq2d 4316 . . . 4  |-  ( y  =  B  ->  (
( z F A ) R ( z F y )  <->  ( z F A ) R ( z F B ) ) )
107, 9imbi12d 320 . . 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 6110 . . . . 5  |-  ( z  =  C  ->  (
z F A )  =  ( C F A ) )
12 oveq1 6110 . . . . 5  |-  ( z  =  C  ->  (
z F B )  =  ( C F B ) )
1311, 12breq12d 4317 . . . 4  |-  ( z  =  C  ->  (
( z F A ) R ( z F B )  <->  ( C F A ) R ( C F B ) ) )
1413imbi2d 316 . . 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 3094 . 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 469 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 setvar class
Syntax hints:    -> wi 4    /\ wa 369    /\ w3a 965    = wceq 1369    e. wcel 1756   A.wral 2727   class class class wbr 4304  (class class class)co 6103
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2577  df-ral 2732  df-rex 2733  df-rab 2736  df-v 2986  df-dif 3343  df-un 3345  df-in 3347  df-ss 3354  df-nul 3650  df-if 3804  df-sn 3890  df-pr 3892  df-op 3896  df-uni 4104  df-br 4305  df-iota 5393  df-fv 5438  df-ov 6106
This theorem is referenced by:  caovordid  6281
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