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Theorem aovnuoveq 31743
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 31670 . . 3  |- (( A F B))  =  ( F''' <. A ,  B >. )
21neeq1i 2752 . 2  |-  ( (( A F B))  =/=  _V  <->  ( F''' <. A ,  B >. )  =/=  _V )
3 afvnufveq 31699 . . 3  |-  ( ( F''' <. A ,  B >. )  =/=  _V  ->  ( F''' <. A ,  B >. )  =  ( F `
 <. A ,  B >. ) )
4 df-ov 6285 . . 3  |-  ( A F B )  =  ( F `  <. A ,  B >. )
53, 1, 43eqtr4g 2533 . 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 1379    =/= wne 2662   _Vcvv 3113   <.cop 4033   ` cfv 5586  (class class class)co 6282  '''cafv 31666   ((caov 31667
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-rab 2823  df-v 3115  df-un 3481  df-if 3940  df-fv 5594  df-ov 6285  df-afv 31669  df-aov 31670
This theorem is referenced by: (None)
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