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Theorem aovnuoveq 32479
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 32406 . . 3  |- (( A F B))  =  ( F''' <. A ,  B >. )
21neeq1i 2742 . 2  |-  ( (( A F B))  =/=  _V  <->  ( F''' <. A ,  B >. )  =/=  _V )
3 afvnufveq 32435 . . 3  |-  ( ( F''' <. A ,  B >. )  =/=  _V  ->  ( F''' <. A ,  B >. )  =  ( F `
 <. A ,  B >. ) )
4 df-ov 6299 . . 3  |-  ( A F B )  =  ( F `  <. A ,  B >. )
53, 1, 43eqtr4g 2523 . 2  |-  ( ( F''' <. A ,  B >. )  =/=  _V  -> (( A F B))  =  ( A F B ) )
62, 5sylbi 195 1  |-  ( (( A F B))  =/=  _V  -> (( A F B))  =  ( A F B ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1395    =/= wne 2652   _Vcvv 3109   <.cop 4038   ` cfv 5594  (class class class)co 6296  '''cafv 32402   ((caov 32403
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-rab 2816  df-v 3111  df-un 3476  df-if 3945  df-fv 5602  df-ov 6299  df-afv 32405  df-aov 32406
This theorem is referenced by: (None)
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