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Theorem aovnuoveq 38563
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 38490 . . 3  |- (( A F B))  =  ( F''' <. A ,  B >. )
21neeq1i 2705 . 2  |-  ( (( A F B))  =/=  _V  <->  ( F''' <. A ,  B >. )  =/=  _V )
3 afvnufveq 38519 . . 3  |-  ( ( F''' <. A ,  B >. )  =/=  _V  ->  ( F''' <. A ,  B >. )  =  ( F `
 <. A ,  B >. ) )
4 df-ov 6308 . . 3  |-  ( A F B )  =  ( F `  <. A ,  B >. )
53, 1, 43eqtr4g 2488 . 2  |-  ( ( F''' <. A ,  B >. )  =/=  _V  -> (( A F B))  =  ( A F B ) )
62, 5sylbi 198 1  |-  ( (( A F B))  =/=  _V  -> (( A F B))  =  ( A F B ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1437    =/= wne 2614   _Vcvv 3080   <.cop 4004   ` cfv 5601  (class class class)co 6305  '''cafv 38486   ((caov 38487
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1663  ax-4 1676  ax-5 1752  ax-6 1798  ax-7 1843  ax-10 1891  ax-11 1896  ax-12 1909  ax-13 2057  ax-ext 2401
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-tru 1440  df-ex 1658  df-nf 1662  df-sb 1791  df-clab 2408  df-cleq 2414  df-clel 2417  df-nfc 2568  df-ne 2616  df-rab 2780  df-v 3082  df-un 3441  df-if 3912  df-fv 5609  df-ov 6308  df-afv 38489  df-aov 38490
This theorem is referenced by: (None)
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