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Theorem aovvoveq 27726
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 27645 . . 3  |- (( A F B))  =  ( F''' <. A ,  B >. )
21eleq1i 2451 . 2  |-  ( (( A F B))  e.  C  <->  ( F''' <. A ,  B >. )  e.  C )
3 afvvfveq 27682 . . 3  |-  ( ( F''' <. A ,  B >. )  e.  C  -> 
( F''' <. A ,  B >. )  =  ( F `
 <. A ,  B >. ) )
4 df-ov 6024 . . 3  |-  ( A F B )  =  ( F `  <. A ,  B >. )
53, 1, 43eqtr4g 2445 . 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 1649    e. wcel 1717   <.cop 3761   ` cfv 5395  (class class class)co 6021  '''cafv 27641   ((caov 27642
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2369  ax-sep 4272
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-clab 2375  df-cleq 2381  df-clel 2384  df-nfc 2513  df-ne 2553  df-rab 2659  df-v 2902  df-un 3269  df-if 3684  df-fv 5403  df-ov 6024  df-afv 27644  df-aov 27645
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