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Theorem caovcang 6248
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 2799 . 2  |-  ( ph  ->  A. x  e.  T  A. y  e.  S  A. z  e.  S  ( ( x F y )  =  ( x F z )  <-> 
y  =  z ) )
3 oveq1 6088 . . . . 5  |-  ( x  =  A  ->  (
x F y )  =  ( A F y ) )
4 oveq1 6088 . . . . 5  |-  ( x  =  A  ->  (
x F z )  =  ( A F z ) )
53, 4eqeq12d 2450 . . . 4  |-  ( x  =  A  ->  (
( x F y )  =  ( x F z )  <->  ( A F y )  =  ( A F z ) ) )
65bibi1d 311 . . 3  |-  ( x  =  A  ->  (
( ( x F y )  =  ( x F z )  <-> 
y  =  z )  <-> 
( ( A F y )  =  ( A F z )  <-> 
y  =  z ) ) )
7 oveq2 6089 . . . . 5  |-  ( y  =  B  ->  ( A F y )  =  ( A F B ) )
87eqeq1d 2444 . . . 4  |-  ( y  =  B  ->  (
( A F y )  =  ( A F z )  <->  ( A F B )  =  ( A F z ) ) )
9 eqeq1 2442 . . . 4  |-  ( y  =  B  ->  (
y  =  z  <->  B  =  z ) )
108, 9bibi12d 313 . . 3  |-  ( y  =  B  ->  (
( ( A F y )  =  ( A F z )  <-> 
y  =  z )  <-> 
( ( A F B )  =  ( A F z )  <-> 
B  =  z ) ) )
11 oveq2 6089 . . . . 5  |-  ( z  =  C  ->  ( A F z )  =  ( A F C ) )
1211eqeq2d 2447 . . . 4  |-  ( z  =  C  ->  (
( A F B )  =  ( A F z )  <->  ( A F B )  =  ( A F C ) ) )
13 eqeq2 2445 . . . 4  |-  ( z  =  C  ->  ( B  =  z  <->  B  =  C ) )
1412, 13bibi12d 313 . . 3  |-  ( z  =  C  ->  (
( ( A F B )  =  ( A F z )  <-> 
B  =  z )  <-> 
( ( A F B )  =  ( A F C )  <-> 
B  =  C ) ) )
156, 10, 14rspc3v 3061 . 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 456 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 177    /\ wa 359    /\ w3a 936    = wceq 1652    e. wcel 1725   A.wral 2705  (class class class)co 6081
This theorem is referenced by:  caovcand  6249  caofcan  27517
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ral 2710  df-rex 2711  df-rab 2714  df-v 2958  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-nul 3629  df-if 3740  df-sn 3820  df-pr 3821  df-op 3823  df-uni 4016  df-br 4213  df-iota 5418  df-fv 5462  df-ov 6084
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