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Theorem aovnuoveq 38784
Description: The alternative value of the operation on an ordered pair equals the operation's value at this ordered pair. (Contributed by Alexander van der Vekens, 26-May-2017.)
Assertion
Ref Expression
aovnuoveq  |-  ( (( A F B))  =/=  _V  -> (( A F B))  =  ( A F B ) )

Proof of Theorem aovnuoveq
StepHypRef Expression
1 df-aov 38710 . . 3  |- (( A F B))  =  ( F''' <. A ,  B >. )
21neeq1i 2688 . 2  |-  ( (( A F B))  =/=  _V  <->  ( F''' <. A ,  B >. )  =/=  _V )
3 afvnufveq 38740 . . 3  |-  ( ( F''' <. A ,  B >. )  =/=  _V  ->  ( F''' <. A ,  B >. )  =  ( F `
 <. A ,  B >. ) )
4 df-ov 6279 . . 3  |-  ( A F B )  =  ( F `  <. A ,  B >. )
53, 1, 43eqtr4g 2511 . 2  |-  ( ( F''' <. A ,  B >. )  =/=  _V  -> (( A F B))  =  ( A F B ) )
62, 5sylbi 200 1  |-  ( (( A F B))  =/=  _V  -> (( A F B))  =  ( A F B ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1448    =/= wne 2622   _Vcvv 3013   <.cop 3942   ` cfv 5561  (class class class)co 6276  '''cafv 38706   ((caov 38707
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1673  ax-4 1686  ax-5 1762  ax-6 1809  ax-7 1855  ax-10 1919  ax-11 1924  ax-12 1937  ax-13 2092  ax-ext 2432
This theorem depends on definitions:  df-bi 190  df-or 376  df-an 377  df-tru 1451  df-ex 1668  df-nf 1672  df-sb 1802  df-clab 2439  df-cleq 2445  df-clel 2448  df-nfc 2582  df-ne 2624  df-rab 2746  df-v 3015  df-un 3377  df-if 3850  df-fv 5569  df-ov 6279  df-afv 38709  df-aov 38710
This theorem is referenced by: (None)
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