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Theorem caovcang 6021
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 2636 . 2  |-  ( ph  ->  A. x  e.  T  A. y  e.  S  A. z  e.  S  ( ( x F y )  =  ( x F z )  <-> 
y  =  z ) )
3 oveq1 5865 . . . . 5  |-  ( x  =  A  ->  (
x F y )  =  ( A F y ) )
4 oveq1 5865 . . . . 5  |-  ( x  =  A  ->  (
x F z )  =  ( A F z ) )
53, 4eqeq12d 2297 . . . 4  |-  ( x  =  A  ->  (
( x F y )  =  ( x F z )  <->  ( A F y )  =  ( A F z ) ) )
65bibi1d 310 . . 3  |-  ( x  =  A  ->  (
( ( x F y )  =  ( x F z )  <-> 
y  =  z )  <-> 
( ( A F y )  =  ( A F z )  <-> 
y  =  z ) ) )
7 oveq2 5866 . . . . 5  |-  ( y  =  B  ->  ( A F y )  =  ( A F B ) )
87eqeq1d 2291 . . . 4  |-  ( y  =  B  ->  (
( A F y )  =  ( A F z )  <->  ( A F B )  =  ( A F z ) ) )
9 eqeq1 2289 . . . 4  |-  ( y  =  B  ->  (
y  =  z  <->  B  =  z ) )
108, 9bibi12d 312 . . 3  |-  ( y  =  B  ->  (
( ( A F y )  =  ( A F z )  <-> 
y  =  z )  <-> 
( ( A F B )  =  ( A F z )  <-> 
B  =  z ) ) )
11 oveq2 5866 . . . . 5  |-  ( z  =  C  ->  ( A F z )  =  ( A F C ) )
1211eqeq2d 2294 . . . 4  |-  ( z  =  C  ->  (
( A F B )  =  ( A F z )  <->  ( A F B )  =  ( A F C ) ) )
13 eqeq2 2292 . . . 4  |-  ( z  =  C  ->  ( B  =  z  <->  B  =  C ) )
1412, 13bibi12d 312 . . 3  |-  ( z  =  C  ->  (
( ( A F B )  =  ( A F z )  <-> 
B  =  z )  <-> 
( ( A F B )  =  ( A F C )  <-> 
B  =  C ) ) )
156, 10, 14rspc3v 2893 . 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 455 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    <-> wb 176    /\ wa 358    /\ w3a 934    = wceq 1623    e. wcel 1684   A.wral 2543  (class class class)co 5858
This theorem is referenced by:  caovcand  6022  caofcan  27540
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-br 4024  df-iota 5219  df-fv 5263  df-ov 5861
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