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Theorem aovnuoveq 27204
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 27124 . . 3  |- (( A F B))  =  ( F''' <. A ,  B >. )
21neeq1i 2489 . 2  |-  ( (( A F B))  =/=  _V  <->  ( F''' <. A ,  B >. )  =/=  _V )
3 afvnufveq 27160 . . 3  |-  ( ( F''' <. A ,  B >. )  =/=  _V  ->  ( F''' <. A ,  B >. )  =  ( F `
 <. A ,  B >. ) )
4 df-ov 5903 . . 3  |-  ( A F B )  =  ( F `  <. A ,  B >. )
53, 1, 43eqtr4g 2373 . 2  |-  ( ( F''' <. A ,  B >. )  =/=  _V  -> (( A F B))  =  ( A F B ) )
62, 5sylbi 187 1  |-  ( (( A F B))  =/=  _V  -> (( A F B))  =  ( A F B ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1633    =/= wne 2479   _Vcvv 2822   <.cop 3677   ` cfv 5292  (class class class)co 5900  '''cafv 27120   ((caov 27121
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1537  ax-5 1548  ax-17 1607  ax-9 1645  ax-8 1666  ax-6 1720  ax-7 1725  ax-11 1732  ax-12 1897  ax-ext 2297
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1310  df-ex 1533  df-nf 1536  df-sb 1640  df-clab 2303  df-cleq 2309  df-clel 2312  df-nfc 2441  df-ne 2481  df-rab 2586  df-v 2824  df-un 3191  df-if 3600  df-fv 5300  df-ov 5903  df-afv 27123  df-aov 27124
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