ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  caovcomg Unicode version

Theorem caovcomg 6006
Description: Convert an operation commutative law to class notation. (Contributed by Mario Carneiro, 1-Jun-2013.)
Hypothesis
Ref Expression
caovcomg.1  |-  ( (
ph  /\  ( x  e.  S  /\  y  e.  S ) )  -> 
( x F y )  =  ( y F x ) )
Assertion
Ref Expression
caovcomg  |-  ( (
ph  /\  ( A  e.  S  /\  B  e.  S ) )  -> 
( A F B )  =  ( B F A ) )
Distinct variable groups:    x, y, A   
x, B, y    ph, x, y    x, F, y    x, S, y

Proof of Theorem caovcomg
StepHypRef Expression
1 caovcomg.1 . . 3  |-  ( (
ph  /\  ( x  e.  S  /\  y  e.  S ) )  -> 
( x F y )  =  ( y F x ) )
21ralrimivva 2552 . 2  |-  ( ph  ->  A. x  e.  S  A. y  e.  S  ( x F y )  =  ( y F x ) )
3 oveq1 5858 . . . 4  |-  ( x  =  A  ->  (
x F y )  =  ( A F y ) )
4 oveq2 5859 . . . 4  |-  ( x  =  A  ->  (
y F x )  =  ( y F A ) )
53, 4eqeq12d 2185 . . 3  |-  ( x  =  A  ->  (
( x F y )  =  ( y F x )  <->  ( A F y )  =  ( y F A ) ) )
6 oveq2 5859 . . . 4  |-  ( y  =  B  ->  ( A F y )  =  ( A F B ) )
7 oveq1 5858 . . . 4  |-  ( y  =  B  ->  (
y F A )  =  ( B F A ) )
86, 7eqeq12d 2185 . . 3  |-  ( y  =  B  ->  (
( A F y )  =  ( y F A )  <->  ( A F B )  =  ( B F A ) ) )
95, 8rspc2v 2847 . 2  |-  ( ( A  e.  S  /\  B  e.  S )  ->  ( A. x  e.  S  A. y  e.  S  ( x F y )  =  ( y F x )  ->  ( A F B )  =  ( B F A ) ) )
102, 9mpan9 279 1  |-  ( (
ph  /\  ( A  e.  S  /\  B  e.  S ) )  -> 
( A F B )  =  ( B F A ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    = wceq 1348    e. wcel 2141   A.wral 2448  (class class class)co 5851
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 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-ext 2152
This theorem depends on definitions:  df-bi 116  df-3an 975  df-tru 1351  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ral 2453  df-rex 2454  df-v 2732  df-un 3125  df-sn 3587  df-pr 3588  df-op 3590  df-uni 3795  df-br 3988  df-iota 5158  df-fv 5204  df-ov 5854
This theorem is referenced by:  caovcomd  6007  caovcom  6008  caovlem2d  6043  caofcom  6082  seq3caopr  10428
  Copyright terms: Public domain W3C validator