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Theorem caovcang 5984
Description: Convert an operation cancellation law to class notation. (Contributed by NM, 20-Aug-1995.) (Revised by Mario Carneiro, 30-Dec-2014.)
Hypothesis
Ref Expression
caovcang.1  |-  ( (
ph  /\  ( x  e.  T  /\  y  e.  S  /\  z  e.  S ) )  -> 
( ( x F y )  =  ( x F z )  <-> 
y  =  z ) )
Assertion
Ref Expression
caovcang  |-  ( (
ph  /\  ( A  e.  T  /\  B  e.  S  /\  C  e.  S ) )  -> 
( ( 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 caovcang
StepHypRef Expression
1 caovcang.1 . . 3  |-  ( (
ph  /\  ( x  e.  T  /\  y  e.  S  /\  z  e.  S ) )  -> 
( ( x F y )  =  ( x F z )  <-> 
y  =  z ) )
21ralrimivvva 2540 . 2  |-  ( ph  ->  A. x  e.  T  A. y  e.  S  A. z  e.  S  ( ( x F y )  =  ( x F z )  <-> 
y  =  z ) )
3 oveq1 5833 . . . . 5  |-  ( x  =  A  ->  (
x F y )  =  ( A F y ) )
4 oveq1 5833 . . . . 5  |-  ( x  =  A  ->  (
x F z )  =  ( A F z ) )
53, 4eqeq12d 2172 . . . 4  |-  ( x  =  A  ->  (
( x F y )  =  ( x F z )  <->  ( A F y )  =  ( A F z ) ) )
65bibi1d 232 . . 3  |-  ( x  =  A  ->  (
( ( x F y )  =  ( x F z )  <-> 
y  =  z )  <-> 
( ( A F y )  =  ( A F z )  <-> 
y  =  z ) ) )
7 oveq2 5834 . . . . 5  |-  ( y  =  B  ->  ( A F y )  =  ( A F B ) )
87eqeq1d 2166 . . . 4  |-  ( y  =  B  ->  (
( A F y )  =  ( A F z )  <->  ( A F B )  =  ( A F z ) ) )
9 eqeq1 2164 . . . 4  |-  ( y  =  B  ->  (
y  =  z  <->  B  =  z ) )
108, 9bibi12d 234 . . 3  |-  ( y  =  B  ->  (
( ( A F y )  =  ( A F z )  <-> 
y  =  z )  <-> 
( ( A F B )  =  ( A F z )  <-> 
B  =  z ) ) )
11 oveq2 5834 . . . . 5  |-  ( z  =  C  ->  ( A F z )  =  ( A F C ) )
1211eqeq2d 2169 . . . 4  |-  ( z  =  C  ->  (
( A F B )  =  ( A F z )  <->  ( A F B )  =  ( A F C ) ) )
13 eqeq2 2167 . . . 4  |-  ( z  =  C  ->  ( B  =  z  <->  B  =  C ) )
1412, 13bibi12d 234 . . 3  |-  ( z  =  C  ->  (
( ( A F B )  =  ( A F z )  <-> 
B  =  z )  <-> 
( ( A F B )  =  ( A F C )  <-> 
B  =  C ) ) )
156, 10, 14rspc3v 2832 . 2  |-  ( ( A  e.  T  /\  B  e.  S  /\  C  e.  S )  ->  ( A. x  e.  T  A. y  e.  S  A. z  e.  S  ( ( x F y )  =  ( x F z )  <->  y  =  z )  ->  ( ( A F B )  =  ( A F C )  <->  B  =  C
) ) )
162, 15mpan9 279 1  |-  ( (
ph  /\  ( A  e.  T  /\  B  e.  S  /\  C  e.  S ) )  -> 
( ( A F B )  =  ( A F C )  <-> 
B  =  C ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    <-> wb 104    /\ w3a 963    = wceq 1335    e. wcel 2128   A.wral 2435  (class class class)co 5826
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2139
This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1338  df-nf 1441  df-sb 1743  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-ral 2440  df-rex 2441  df-v 2714  df-un 3106  df-sn 3567  df-pr 3568  df-op 3570  df-uni 3775  df-br 3968  df-iota 5137  df-fv 5180  df-ov 5829
This theorem is referenced by:  caovcand  5985
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