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Theorem aovnuoveq 46634
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 ( ((𝐴𝐹𝐵)) ≠ V → ((𝐴𝐹𝐵)) = (𝐴𝐹𝐵))

Proof of Theorem aovnuoveq
StepHypRef Expression
1 df-aov 46564 . . 3 ((𝐴𝐹𝐵)) = (𝐹'''⟨𝐴, 𝐵⟩)
21neeq1i 2995 . 2 ( ((𝐴𝐹𝐵)) ≠ V ↔ (𝐹'''⟨𝐴, 𝐵⟩) ≠ V)
3 afvnufveq 46590 . . 3 ((𝐹'''⟨𝐴, 𝐵⟩) ≠ V → (𝐹'''⟨𝐴, 𝐵⟩) = (𝐹‘⟨𝐴, 𝐵⟩))
4 df-ov 7419 . . 3 (𝐴𝐹𝐵) = (𝐹‘⟨𝐴, 𝐵⟩)
53, 1, 43eqtr4g 2790 . 2 ((𝐹'''⟨𝐴, 𝐵⟩) ≠ V → ((𝐴𝐹𝐵)) = (𝐴𝐹𝐵))
62, 5sylbi 216 1 ( ((𝐴𝐹𝐵)) ≠ V → ((𝐴𝐹𝐵)) = (𝐴𝐹𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1533  wne 2930  Vcvv 3463  cop 4630  cfv 6543  (class class class)co 7416  '''cafv 46560   ((caov 46561
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2696  ax-sep 5294  ax-nul 5301  ax-pr 5423
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2703  df-cleq 2717  df-clel 2802  df-nfc 2877  df-ne 2931  df-ral 3052  df-rex 3061  df-rab 3420  df-v 3465  df-sbc 3769  df-csb 3885  df-dif 3942  df-un 3944  df-in 3946  df-ss 3956  df-nul 4319  df-if 4525  df-sn 4625  df-pr 4627  df-op 4631  df-uni 4904  df-int 4945  df-br 5144  df-opab 5206  df-id 5570  df-xp 5678  df-rel 5679  df-cnv 5680  df-co 5681  df-dm 5682  df-res 5684  df-iota 6495  df-fun 6545  df-fv 6551  df-ov 7419  df-aiota 46528  df-dfat 46562  df-afv 46563  df-aov 46564
This theorem is referenced by: (None)
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