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Theorem aovvoveq 28013
Description: The alternative value of the operation on an ordered pair equals the operation's value on this ordered pair. (Contributed by Alexander van der Vekens, 26-May-2017.)
Assertion
Ref Expression
aovvoveq  |-  ( (( A F B))  e.  C  -> (( A F B))  =  ( A F B ) )

Proof of Theorem aovvoveq
StepHypRef Expression
1 df-aov 27933 . . 3  |- (( A F B))  =  ( F''' <. A ,  B >. )
21eleq1i 2498 . 2  |-  ( (( A F B))  e.  C  <->  ( F''' <. A ,  B >. )  e.  C )
3 afvvfveq 27969 . . 3  |-  ( ( F''' <. A ,  B >. )  e.  C  -> 
( F''' <. A ,  B >. )  =  ( F `
 <. A ,  B >. ) )
4 df-ov 6076 . . 3  |-  ( A F B )  =  ( F `  <. A ,  B >. )
53, 1, 43eqtr4g 2492 . 2  |-  ( ( F''' <. A ,  B >. )  e.  C  -> (( A F B))  =  ( A F B ) )
62, 5sylbi 188 1  |-  ( (( A F B))  e.  C  -> (( A F B))  =  ( A F B ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1652    e. wcel 1725   <.cop 3809   ` cfv 5446  (class class class)co 6073  '''cafv 27929   ((caov 27930
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-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-sep 4322
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-rab 2706  df-v 2950  df-un 3317  df-if 3732  df-fv 5454  df-ov 6076  df-afv 27932  df-aov 27933
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