Theorem aovnuoveq 41271

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

Step	Hyp	Ref	Expression
1		df-aov 41198	. . 3 ⊢ ((𝐴𝐹𝐵)) = (𝐹'''⟨𝐴, 𝐵⟩)
2	1	neeq1i 2858	. 2 ⊢ ( ((𝐴𝐹𝐵)) ≠ V ↔ (𝐹'''⟨𝐴, 𝐵⟩) ≠ V)
3		afvnufveq 41227	. . 3 ⊢ ((𝐹'''⟨𝐴, 𝐵⟩) ≠ V → (𝐹'''⟨𝐴, 𝐵⟩) = (𝐹‘⟨𝐴, 𝐵⟩))
4		df-ov 6653	. . 3 ⊢ (𝐴𝐹𝐵) = (𝐹‘⟨𝐴, 𝐵⟩)
5	3, 1, 4	3eqtr4g 2681	. 2 ⊢ ((𝐹'''⟨𝐴, 𝐵⟩) ≠ V → ((𝐴𝐹𝐵)) = (𝐴𝐹𝐵))
6	2, 5	sylbi 207	1 ⊢ ( ((𝐴𝐹𝐵)) ≠ V → ((𝐴𝐹𝐵)) = (𝐴𝐹𝐵))

Syntax hints:   → wi 4   = wceq 1483   ≠ wne 2794  Vcvv 3200  ⟨cop 4183  ‘cfv 5888  (class class class)co 6650  '''cafv 41194   ((caov 41195

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-rab 2921  df-v 3202  df-un 3579  df-if 4087  df-fv 5896  df-ov 6653  df-afv 41197  df-aov 41198

This theorem is referenced by: (None)