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Theorem caovcand 6181
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 20 . 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 6180 . 2  |-  ( (
ph  /\  ( A  e.  T  /\  B  e.  S  /\  C  e.  S ) )  -> 
( ( A F B )  =  ( A F C )  <-> 
B  =  C ) )
71, 2, 3, 4, 6syl13anc 1186 1  |-  ( ph  ->  ( ( A F B )  =  ( A F C )  <-> 
B  =  C ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359    /\ w3a 936    = wceq 1649    e. wcel 1717  (class class class)co 6013
This theorem is referenced by:  caovcanrd  6182
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2361
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-clab 2367  df-cleq 2373  df-clel 2376  df-nfc 2505  df-ral 2647  df-rex 2648  df-rab 2651  df-v 2894  df-dif 3259  df-un 3261  df-in 3263  df-ss 3270  df-nul 3565  df-if 3676  df-sn 3756  df-pr 3757  df-op 3759  df-uni 3951  df-br 4147  df-iota 5351  df-fv 5395  df-ov 6016
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