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Theorem aovnuoveq 45497
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 45427 . . 3 ((𝐴𝐹𝐵)) = (𝐹'''⟨𝐴, 𝐵⟩)
21neeq1i 3009 . 2 ( ((𝐴𝐹𝐵)) ≠ V ↔ (𝐹'''⟨𝐴, 𝐵⟩) ≠ V)
3 afvnufveq 45453 . . 3 ((𝐹'''⟨𝐴, 𝐵⟩) ≠ V → (𝐹'''⟨𝐴, 𝐵⟩) = (𝐹‘⟨𝐴, 𝐵⟩))
4 df-ov 7365 . . 3 (𝐴𝐹𝐵) = (𝐹‘⟨𝐴, 𝐵⟩)
53, 1, 43eqtr4g 2802 . 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 1542  wne 2944  Vcvv 3448  cop 4597  cfv 6501  (class class class)co 7362  '''cafv 45423   ((caov 45424
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2708  ax-sep 5261  ax-nul 5268  ax-pr 5389
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2815  df-nfc 2890  df-ne 2945  df-ral 3066  df-rex 3075  df-rab 3411  df-v 3450  df-sbc 3745  df-csb 3861  df-dif 3918  df-un 3920  df-in 3922  df-ss 3932  df-nul 4288  df-if 4492  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4871  df-int 4913  df-br 5111  df-opab 5173  df-id 5536  df-xp 5644  df-rel 5645  df-cnv 5646  df-co 5647  df-dm 5648  df-res 5650  df-iota 6453  df-fun 6503  df-fv 6509  df-ov 7365  df-aiota 45391  df-dfat 45425  df-afv 45426  df-aov 45427
This theorem is referenced by: (None)
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