MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  caovcang Structured version   Unicode version

Theorem caovcang 6457
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 2863 . 2  |-  ( ph  ->  A. x  e.  T  A. y  e.  S  A. z  e.  S  ( ( x F y )  =  ( x F z )  <-> 
y  =  z ) )
3 oveq1 6284 . . . . 5  |-  ( x  =  A  ->  (
x F y )  =  ( A F y ) )
4 oveq1 6284 . . . . 5  |-  ( x  =  A  ->  (
x F z )  =  ( A F z ) )
53, 4eqeq12d 2463 . . . 4  |-  ( x  =  A  ->  (
( x F y )  =  ( x F z )  <->  ( A F y )  =  ( A F z ) ) )
65bibi1d 319 . . 3  |-  ( x  =  A  ->  (
( ( x F y )  =  ( x F z )  <-> 
y  =  z )  <-> 
( ( A F y )  =  ( A F z )  <-> 
y  =  z ) ) )
7 oveq2 6285 . . . . 5  |-  ( y  =  B  ->  ( A F y )  =  ( A F B ) )
87eqeq1d 2443 . . . 4  |-  ( y  =  B  ->  (
( A F y )  =  ( A F z )  <->  ( A F B )  =  ( A F z ) ) )
9 eqeq1 2445 . . . 4  |-  ( y  =  B  ->  (
y  =  z  <->  B  =  z ) )
108, 9bibi12d 321 . . 3  |-  ( y  =  B  ->  (
( ( A F y )  =  ( A F z )  <-> 
y  =  z )  <-> 
( ( A F B )  =  ( A F z )  <-> 
B  =  z ) ) )
11 oveq2 6285 . . . . 5  |-  ( z  =  C  ->  ( A F z )  =  ( A F C ) )
1211eqeq2d 2455 . . . 4  |-  ( z  =  C  ->  (
( A F B )  =  ( A F z )  <->  ( A F B )  =  ( A F C ) ) )
13 eqeq2 2456 . . . 4  |-  ( z  =  C  ->  ( B  =  z  <->  B  =  C ) )
1412, 13bibi12d 321 . . 3  |-  ( z  =  C  ->  (
( ( A F B )  =  ( A F z )  <-> 
B  =  z )  <-> 
( ( A F B )  =  ( A F C )  <-> 
B  =  C ) ) )
156, 10, 14rspc3v 3206 . 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 469 1  |-  ( (
ph  /\  ( A  e.  T  /\  B  e.  S  /\  C  e.  S ) )  -> 
( ( A F B )  =  ( A F C )  <-> 
B  =  C ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 972    = wceq 1381    e. wcel 1802   A.wral 2791  (class class class)co 6277
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1603  ax-4 1616  ax-5 1689  ax-6 1732  ax-7 1774  ax-10 1821  ax-11 1826  ax-12 1838  ax-13 1983  ax-ext 2419
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 974  df-tru 1384  df-ex 1598  df-nf 1602  df-sb 1725  df-clab 2427  df-cleq 2433  df-clel 2436  df-nfc 2591  df-ral 2796  df-rex 2797  df-rab 2800  df-v 3095  df-dif 3461  df-un 3463  df-in 3465  df-ss 3472  df-nul 3768  df-if 3923  df-sn 4011  df-pr 4013  df-op 4017  df-uni 4231  df-br 4434  df-iota 5537  df-fv 5582  df-ov 6280
This theorem is referenced by:  caovcand  6458  caofcan  31197
  Copyright terms: Public domain W3C validator