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Theorem caovcand 6038
Description: Convert an operation cancellation law to class notation. (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 )
Assertion
Ref Expression
caovcand  |-  ( ph  ->  ( ( A F B )  =  ( A F C )  <-> 
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 caovcand
StepHypRef Expression
1 id 19 . 2  |-  ( ph  ->  ph )
2 caovcand.2 . 2  |-  ( ph  ->  A  e.  T )
3 caovcand.3 . 2  |-  ( ph  ->  B  e.  S )
4 caovcand.4 . 2  |-  ( ph  ->  C  e.  S )
5 caovcang.1 . . 3  |-  ( (
ph  /\  ( x  e.  T  /\  y  e.  S  /\  z  e.  S ) )  -> 
( ( x F y )  =  ( x F z )  <-> 
y  =  z ) )
65caovcang 6037 . 2  |-  ( (
ph  /\  ( A  e.  T  /\  B  e.  S  /\  C  e.  S ) )  -> 
( ( A F B )  =  ( A F C )  <-> 
B  =  C ) )
71, 2, 3, 4, 6syl13anc 1184 1  |-  ( ph  ->  ( ( A F B )  =  ( A F C )  <-> 
B  =  C ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    /\ w3a 934    = wceq 1632    e. wcel 1696  (class class class)co 5874
This theorem is referenced by:  caovcanrd  6039
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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