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Theorem caovcanrd 6264
Description: Commute the arguments of an operation cancellation law. (Contributed by Mario Carneiro, 30-Dec-2014.)
Hypotheses
Ref Expression
caovcang.1  |-  ( (
ph  /\  ( x  e.  T  /\  y  e.  S  /\  z  e.  S ) )  -> 
( ( x F y )  =  ( x F z )  <-> 
y  =  z ) )
caovcand.2  |-  ( ph  ->  A  e.  T )
caovcand.3  |-  ( ph  ->  B  e.  S )
caovcand.4  |-  ( ph  ->  C  e.  S )
caovcanrd.5  |-  ( ph  ->  A  e.  S )
caovcanrd.6  |-  ( (
ph  /\  ( x  e.  S  /\  y  e.  S ) )  -> 
( x F y )  =  ( y F x ) )
Assertion
Ref Expression
caovcanrd  |-  ( ph  ->  ( ( B F A )  =  ( C F A )  <-> 
B  =  C ) )
Distinct variable groups:    x, y,
z, A    x, B, y, z    x, C, y, z    ph, x, y, z   
x, F, y, z   
x, S, y, z   
x, T, y, z

Proof of Theorem caovcanrd
StepHypRef Expression
1 caovcanrd.6 . . . 4  |-  ( (
ph  /\  ( x  e.  S  /\  y  e.  S ) )  -> 
( x F y )  =  ( y F x ) )
2 caovcanrd.5 . . . 4  |-  ( ph  ->  A  e.  S )
3 caovcand.3 . . . 4  |-  ( ph  ->  B  e.  S )
41, 2, 3caovcomd 6257 . . 3  |-  ( ph  ->  ( A F B )  =  ( B F A ) )
5 caovcand.4 . . . 4  |-  ( ph  ->  C  e.  S )
61, 2, 5caovcomd 6257 . . 3  |-  ( ph  ->  ( A F C )  =  ( C F A ) )
74, 6eqeq12d 2455 . 2  |-  ( ph  ->  ( ( A F B )  =  ( A F C )  <-> 
( B F A )  =  ( C F A ) ) )
8 caovcang.1 . . 3  |-  ( (
ph  /\  ( x  e.  T  /\  y  e.  S  /\  z  e.  S ) )  -> 
( ( x F y )  =  ( x F z )  <-> 
y  =  z ) )
9 caovcand.2 . . 3  |-  ( ph  ->  A  e.  T )
108, 9, 3, 5caovcand 6263 . 2  |-  ( ph  ->  ( ( A F B )  =  ( A F C )  <-> 
B  =  C ) )
117, 10bitr3d 255 1  |-  ( ph  ->  ( ( B F A )  =  ( C F A )  <-> 
B  =  C ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 965    = wceq 1369    e. wcel 1756  (class class class)co 6089
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 2422
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 2428  df-cleq 2434  df-clel 2437  df-nfc 2566  df-ral 2718  df-rex 2719  df-rab 2722  df-v 2972  df-dif 3329  df-un 3331  df-in 3333  df-ss 3340  df-nul 3636  df-if 3790  df-sn 3876  df-pr 3878  df-op 3882  df-uni 4090  df-br 4291  df-iota 5379  df-fv 5424  df-ov 6092
This theorem is referenced by: (None)
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