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Theorem aovvoveq 28160
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 28079 . . 3  |- (( A F B))  =  ( F''' <. A ,  B >. )
21eleq1i 2359 . 2  |-  ( (( A F B))  e.  C  <->  ( F''' <. A ,  B >. )  e.  C )
3 afvvfveq 28116 . . 3  |-  ( ( F''' <. A ,  B >. )  e.  C  -> 
( F''' <. A ,  B >. )  =  ( F `
 <. A ,  B >. ) )
4 df-ov 5877 . . 3  |-  ( A F B )  =  ( F `  <. A ,  B >. )
53, 1, 43eqtr4g 2353 . 2  |-  ( ( F''' <. A ,  B >. )  e.  C  -> (( A F B))  =  ( A F B ) )
62, 5sylbi 187 1  |-  ( (( A F B))  e.  C  -> (( A F B))  =  ( A F B ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1632    e. wcel 1696   <.cop 3656   ` cfv 5271  (class class class)co 5874  '''cafv 28075   ((caov 28076
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-rab 2565  df-v 2803  df-un 3170  df-if 3579  df-fv 5279  df-ov 5877  df-afv 28078  df-aov 28079
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