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Theorem gonafv 35335
Description: The "Godel-set for the Sheffer stroke NAND" for two formulas 𝐴 and 𝐵. (Contributed by AV, 16-Oct-2023.)
Assertion
Ref Expression
gonafv ((𝐴𝑉𝐵𝑊) → (𝐴𝑔𝐵) = ⟨1o, ⟨𝐴, 𝐵⟩⟩)

Proof of Theorem gonafv
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 df-ov 7434 . 2 (𝐴𝑔𝐵) = (⊼𝑔‘⟨𝐴, 𝐵⟩)
2 opelvvg 5730 . . 3 ((𝐴𝑉𝐵𝑊) → ⟨𝐴, 𝐵⟩ ∈ (V × V))
3 opeq2 4879 . . . 4 (𝑥 = ⟨𝐴, 𝐵⟩ → ⟨1o, 𝑥⟩ = ⟨1o, ⟨𝐴, 𝐵⟩⟩)
4 df-gona 35326 . . . 4 𝑔 = (𝑥 ∈ (V × V) ↦ ⟨1o, 𝑥⟩)
5 opex 5475 . . . 4 ⟨1o, ⟨𝐴, 𝐵⟩⟩ ∈ V
63, 4, 5fvmpt 7016 . . 3 (⟨𝐴, 𝐵⟩ ∈ (V × V) → (⊼𝑔‘⟨𝐴, 𝐵⟩) = ⟨1o, ⟨𝐴, 𝐵⟩⟩)
72, 6syl 17 . 2 ((𝐴𝑉𝐵𝑊) → (⊼𝑔‘⟨𝐴, 𝐵⟩) = ⟨1o, ⟨𝐴, 𝐵⟩⟩)
81, 7eqtrid 2787 1 ((𝐴𝑉𝐵𝑊) → (𝐴𝑔𝐵) = ⟨1o, ⟨𝐴, 𝐵⟩⟩)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1537  wcel 2106  Vcvv 3478  cop 4637   × cxp 5687  cfv 6563  (class class class)co 7431  1oc1o 8498  𝑔cgna 35319
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pr 5438
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-ss 3980  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-iota 6516  df-fun 6565  df-fv 6571  df-ov 7434  df-gona 35326
This theorem is referenced by:  gonanegoal  35337  fmlaomn0  35375  gonan0  35377  gonarlem  35379  gonar  35380  fmla0disjsuc  35383  fmlasucdisj  35384  satffunlem  35386  satffunlem1lem1  35387  satffunlem2lem1  35389
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