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Theorem gonafv 32601
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 7162 . 2 (𝐴𝑔𝐵) = (⊼𝑔‘⟨𝐴, 𝐵⟩)
2 opelvvg 5598 . . 3 ((𝐴𝑉𝐵𝑊) → ⟨𝐴, 𝐵⟩ ∈ (V × V))
3 opeq2 4807 . . . 4 (𝑥 = ⟨𝐴, 𝐵⟩ → ⟨1o, 𝑥⟩ = ⟨1o, ⟨𝐴, 𝐵⟩⟩)
4 df-gona 32592 . . . 4 𝑔 = (𝑥 ∈ (V × V) ↦ ⟨1o, 𝑥⟩)
5 opex 5359 . . . 4 ⟨1o, ⟨𝐴, 𝐵⟩⟩ ∈ V
63, 4, 5fvmpt 6771 . . 3 (⟨𝐴, 𝐵⟩ ∈ (V × V) → (⊼𝑔‘⟨𝐴, 𝐵⟩) = ⟨1o, ⟨𝐴, 𝐵⟩⟩)
72, 6syl 17 . 2 ((𝐴𝑉𝐵𝑊) → (⊼𝑔‘⟨𝐴, 𝐵⟩) = ⟨1o, ⟨𝐴, 𝐵⟩⟩)
81, 7syl5eq 2871 1 ((𝐴𝑉𝐵𝑊) → (𝐴𝑔𝐵) = ⟨1o, ⟨𝐴, 𝐵⟩⟩)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 398   = wceq 1536  wcel 2113  Vcvv 3497  cop 4576   × cxp 5556  cfv 6358  (class class class)co 7159  1oc1o 8098  𝑔cgna 32585
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1969  ax-7 2014  ax-8 2115  ax-9 2123  ax-10 2144  ax-11 2160  ax-12 2176  ax-ext 2796  ax-sep 5206  ax-nul 5213  ax-pr 5333
This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1539  df-ex 1780  df-nf 1784  df-sb 2069  df-mo 2621  df-eu 2653  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2966  df-ral 3146  df-rex 3147  df-rab 3150  df-v 3499  df-sbc 3776  df-dif 3942  df-un 3944  df-in 3946  df-ss 3955  df-nul 4295  df-if 4471  df-sn 4571  df-pr 4573  df-op 4577  df-uni 4842  df-br 5070  df-opab 5132  df-mpt 5150  df-id 5463  df-xp 5564  df-rel 5565  df-cnv 5566  df-co 5567  df-dm 5568  df-iota 6317  df-fun 6360  df-fv 6366  df-ov 7162  df-gona 32592
This theorem is referenced by:  gonanegoal  32603  fmlaomn0  32641  gonan0  32643  gonarlem  32645  gonar  32646  fmla0disjsuc  32649  fmlasucdisj  32650  satffunlem  32652  satffunlem1lem1  32653  satffunlem2lem1  32655
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