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Theorem 2goelgoanfmla1 34018
Description: Two Godel-sets of membership combined with a Godel-set for NAND is a Godel formula of height 1. (Contributed by AV, 17-Nov-2023.)
Hypothesis
Ref Expression
satfv1fvfmla1.x 𝑋 = ((𝐼𝑔𝐽)⊼𝑔(𝐾𝑔𝐿))
Assertion
Ref Expression
2goelgoanfmla1 (((𝐼 ∈ ω ∧ 𝐽 ∈ ω) ∧ (𝐾 ∈ ω ∧ 𝐿 ∈ ω)) → 𝑋 ∈ (Fmla‘1o))

Proof of Theorem 2goelgoanfmla1
Dummy variables 𝑖 𝑗 𝑘 𝑛 𝑥 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 simpll 765 . . . . . 6 (((𝐼 ∈ ω ∧ 𝐽 ∈ ω) ∧ (𝐾 ∈ ω ∧ 𝐿 ∈ ω)) → 𝐼 ∈ ω)
2 simplr 767 . . . . . 6 (((𝐼 ∈ ω ∧ 𝐽 ∈ ω) ∧ (𝐾 ∈ ω ∧ 𝐿 ∈ ω)) → 𝐽 ∈ ω)
3 simprl 769 . . . . . 6 (((𝐼 ∈ ω ∧ 𝐽 ∈ ω) ∧ (𝐾 ∈ ω ∧ 𝐿 ∈ ω)) → 𝐾 ∈ ω)
4 simprr 771 . . . . . . . 8 (((𝐼 ∈ ω ∧ 𝐽 ∈ ω) ∧ (𝐾 ∈ ω ∧ 𝐿 ∈ ω)) → 𝐿 ∈ ω)
5 oveq2 7365 . . . . . . . . . . 11 (𝑛 = 𝐿 → (𝐾𝑔𝑛) = (𝐾𝑔𝐿))
65oveq2d 7373 . . . . . . . . . 10 (𝑛 = 𝐿 → ((𝐼𝑔𝐽)⊼𝑔(𝐾𝑔𝑛)) = ((𝐼𝑔𝐽)⊼𝑔(𝐾𝑔𝐿)))
76eqeq2d 2747 . . . . . . . . 9 (𝑛 = 𝐿 → (𝑋 = ((𝐼𝑔𝐽)⊼𝑔(𝐾𝑔𝑛)) ↔ 𝑋 = ((𝐼𝑔𝐽)⊼𝑔(𝐾𝑔𝐿))))
87adantl 482 . . . . . . . 8 ((((𝐼 ∈ ω ∧ 𝐽 ∈ ω) ∧ (𝐾 ∈ ω ∧ 𝐿 ∈ ω)) ∧ 𝑛 = 𝐿) → (𝑋 = ((𝐼𝑔𝐽)⊼𝑔(𝐾𝑔𝑛)) ↔ 𝑋 = ((𝐼𝑔𝐽)⊼𝑔(𝐾𝑔𝐿))))
9 satfv1fvfmla1.x . . . . . . . . 9 𝑋 = ((𝐼𝑔𝐽)⊼𝑔(𝐾𝑔𝐿))
109a1i 11 . . . . . . . 8 (((𝐼 ∈ ω ∧ 𝐽 ∈ ω) ∧ (𝐾 ∈ ω ∧ 𝐿 ∈ ω)) → 𝑋 = ((𝐼𝑔𝐽)⊼𝑔(𝐾𝑔𝐿)))
114, 8, 10rspcedvd 3583 . . . . . . 7 (((𝐼 ∈ ω ∧ 𝐽 ∈ ω) ∧ (𝐾 ∈ ω ∧ 𝐿 ∈ ω)) → ∃𝑛 ∈ ω 𝑋 = ((𝐼𝑔𝐽)⊼𝑔(𝐾𝑔𝑛)))
1211orcd 871 . . . . . 6 (((𝐼 ∈ ω ∧ 𝐽 ∈ ω) ∧ (𝐾 ∈ ω ∧ 𝐿 ∈ ω)) → (∃𝑛 ∈ ω 𝑋 = ((𝐼𝑔𝐽)⊼𝑔(𝐾𝑔𝑛)) ∨ 𝑋 = ∀𝑔𝐾(𝐼𝑔𝐽)))
13 oveq1 7364 . . . . . . . . . . 11 (𝑖 = 𝐼 → (𝑖𝑔𝑗) = (𝐼𝑔𝑗))
1413oveq1d 7372 . . . . . . . . . 10 (𝑖 = 𝐼 → ((𝑖𝑔𝑗)⊼𝑔(𝑘𝑔𝑛)) = ((𝐼𝑔𝑗)⊼𝑔(𝑘𝑔𝑛)))
1514eqeq2d 2747 . . . . . . . . 9 (𝑖 = 𝐼 → (𝑋 = ((𝑖𝑔𝑗)⊼𝑔(𝑘𝑔𝑛)) ↔ 𝑋 = ((𝐼𝑔𝑗)⊼𝑔(𝑘𝑔𝑛))))
1615rexbidv 3175 . . . . . . . 8 (𝑖 = 𝐼 → (∃𝑛 ∈ ω 𝑋 = ((𝑖𝑔𝑗)⊼𝑔(𝑘𝑔𝑛)) ↔ ∃𝑛 ∈ ω 𝑋 = ((𝐼𝑔𝑗)⊼𝑔(𝑘𝑔𝑛))))
17 eqidd 2737 . . . . . . . . . 10 (𝑖 = 𝐼𝑘 = 𝑘)
1817, 13goaleq12d 33945 . . . . . . . . 9 (𝑖 = 𝐼 → ∀𝑔𝑘(𝑖𝑔𝑗) = ∀𝑔𝑘(𝐼𝑔𝑗))
1918eqeq2d 2747 . . . . . . . 8 (𝑖 = 𝐼 → (𝑋 = ∀𝑔𝑘(𝑖𝑔𝑗) ↔ 𝑋 = ∀𝑔𝑘(𝐼𝑔𝑗)))
2016, 19orbi12d 917 . . . . . . 7 (𝑖 = 𝐼 → ((∃𝑛 ∈ ω 𝑋 = ((𝑖𝑔𝑗)⊼𝑔(𝑘𝑔𝑛)) ∨ 𝑋 = ∀𝑔𝑘(𝑖𝑔𝑗)) ↔ (∃𝑛 ∈ ω 𝑋 = ((𝐼𝑔𝑗)⊼𝑔(𝑘𝑔𝑛)) ∨ 𝑋 = ∀𝑔𝑘(𝐼𝑔𝑗))))
21 oveq2 7365 . . . . . . . . . . 11 (𝑗 = 𝐽 → (𝐼𝑔𝑗) = (𝐼𝑔𝐽))
2221oveq1d 7372 . . . . . . . . . 10 (𝑗 = 𝐽 → ((𝐼𝑔𝑗)⊼𝑔(𝑘𝑔𝑛)) = ((𝐼𝑔𝐽)⊼𝑔(𝑘𝑔𝑛)))
2322eqeq2d 2747 . . . . . . . . 9 (𝑗 = 𝐽 → (𝑋 = ((𝐼𝑔𝑗)⊼𝑔(𝑘𝑔𝑛)) ↔ 𝑋 = ((𝐼𝑔𝐽)⊼𝑔(𝑘𝑔𝑛))))
2423rexbidv 3175 . . . . . . . 8 (𝑗 = 𝐽 → (∃𝑛 ∈ ω 𝑋 = ((𝐼𝑔𝑗)⊼𝑔(𝑘𝑔𝑛)) ↔ ∃𝑛 ∈ ω 𝑋 = ((𝐼𝑔𝐽)⊼𝑔(𝑘𝑔𝑛))))
25 eqidd 2737 . . . . . . . . . 10 (𝑗 = 𝐽𝑘 = 𝑘)
2625, 21goaleq12d 33945 . . . . . . . . 9 (𝑗 = 𝐽 → ∀𝑔𝑘(𝐼𝑔𝑗) = ∀𝑔𝑘(𝐼𝑔𝐽))
2726eqeq2d 2747 . . . . . . . 8 (𝑗 = 𝐽 → (𝑋 = ∀𝑔𝑘(𝐼𝑔𝑗) ↔ 𝑋 = ∀𝑔𝑘(𝐼𝑔𝐽)))
2824, 27orbi12d 917 . . . . . . 7 (𝑗 = 𝐽 → ((∃𝑛 ∈ ω 𝑋 = ((𝐼𝑔𝑗)⊼𝑔(𝑘𝑔𝑛)) ∨ 𝑋 = ∀𝑔𝑘(𝐼𝑔𝑗)) ↔ (∃𝑛 ∈ ω 𝑋 = ((𝐼𝑔𝐽)⊼𝑔(𝑘𝑔𝑛)) ∨ 𝑋 = ∀𝑔𝑘(𝐼𝑔𝐽))))
29 oveq1 7364 . . . . . . . . . . 11 (𝑘 = 𝐾 → (𝑘𝑔𝑛) = (𝐾𝑔𝑛))
3029oveq2d 7373 . . . . . . . . . 10 (𝑘 = 𝐾 → ((𝐼𝑔𝐽)⊼𝑔(𝑘𝑔𝑛)) = ((𝐼𝑔𝐽)⊼𝑔(𝐾𝑔𝑛)))
3130eqeq2d 2747 . . . . . . . . 9 (𝑘 = 𝐾 → (𝑋 = ((𝐼𝑔𝐽)⊼𝑔(𝑘𝑔𝑛)) ↔ 𝑋 = ((𝐼𝑔𝐽)⊼𝑔(𝐾𝑔𝑛))))
3231rexbidv 3175 . . . . . . . 8 (𝑘 = 𝐾 → (∃𝑛 ∈ ω 𝑋 = ((𝐼𝑔𝐽)⊼𝑔(𝑘𝑔𝑛)) ↔ ∃𝑛 ∈ ω 𝑋 = ((𝐼𝑔𝐽)⊼𝑔(𝐾𝑔𝑛))))
33 id 22 . . . . . . . . . 10 (𝑘 = 𝐾𝑘 = 𝐾)
34 eqidd 2737 . . . . . . . . . 10 (𝑘 = 𝐾 → (𝐼𝑔𝐽) = (𝐼𝑔𝐽))
3533, 34goaleq12d 33945 . . . . . . . . 9 (𝑘 = 𝐾 → ∀𝑔𝑘(𝐼𝑔𝐽) = ∀𝑔𝐾(𝐼𝑔𝐽))
3635eqeq2d 2747 . . . . . . . 8 (𝑘 = 𝐾 → (𝑋 = ∀𝑔𝑘(𝐼𝑔𝐽) ↔ 𝑋 = ∀𝑔𝐾(𝐼𝑔𝐽)))
3732, 36orbi12d 917 . . . . . . 7 (𝑘 = 𝐾 → ((∃𝑛 ∈ ω 𝑋 = ((𝐼𝑔𝐽)⊼𝑔(𝑘𝑔𝑛)) ∨ 𝑋 = ∀𝑔𝑘(𝐼𝑔𝐽)) ↔ (∃𝑛 ∈ ω 𝑋 = ((𝐼𝑔𝐽)⊼𝑔(𝐾𝑔𝑛)) ∨ 𝑋 = ∀𝑔𝐾(𝐼𝑔𝐽))))
3820, 28, 37rspc3ev 3594 . . . . . 6 (((𝐼 ∈ ω ∧ 𝐽 ∈ ω ∧ 𝐾 ∈ ω) ∧ (∃𝑛 ∈ ω 𝑋 = ((𝐼𝑔𝐽)⊼𝑔(𝐾𝑔𝑛)) ∨ 𝑋 = ∀𝑔𝐾(𝐼𝑔𝐽))) → ∃𝑖 ∈ ω ∃𝑗 ∈ ω ∃𝑘 ∈ ω (∃𝑛 ∈ ω 𝑋 = ((𝑖𝑔𝑗)⊼𝑔(𝑘𝑔𝑛)) ∨ 𝑋 = ∀𝑔𝑘(𝑖𝑔𝑗)))
391, 2, 3, 12, 38syl31anc 1373 . . . . 5 (((𝐼 ∈ ω ∧ 𝐽 ∈ ω) ∧ (𝐾 ∈ ω ∧ 𝐿 ∈ ω)) → ∃𝑖 ∈ ω ∃𝑗 ∈ ω ∃𝑘 ∈ ω (∃𝑛 ∈ ω 𝑋 = ((𝑖𝑔𝑗)⊼𝑔(𝑘𝑔𝑛)) ∨ 𝑋 = ∀𝑔𝑘(𝑖𝑔𝑗)))
409ovexi 7391 . . . . . 6 𝑋 ∈ V
41 eqeq1 2740 . . . . . . . . . 10 (𝑥 = 𝑋 → (𝑥 = ((𝑖𝑔𝑗)⊼𝑔(𝑘𝑔𝑛)) ↔ 𝑋 = ((𝑖𝑔𝑗)⊼𝑔(𝑘𝑔𝑛))))
4241rexbidv 3175 . . . . . . . . 9 (𝑥 = 𝑋 → (∃𝑛 ∈ ω 𝑥 = ((𝑖𝑔𝑗)⊼𝑔(𝑘𝑔𝑛)) ↔ ∃𝑛 ∈ ω 𝑋 = ((𝑖𝑔𝑗)⊼𝑔(𝑘𝑔𝑛))))
43 eqeq1 2740 . . . . . . . . 9 (𝑥 = 𝑋 → (𝑥 = ∀𝑔𝑘(𝑖𝑔𝑗) ↔ 𝑋 = ∀𝑔𝑘(𝑖𝑔𝑗)))
4442, 43orbi12d 917 . . . . . . . 8 (𝑥 = 𝑋 → ((∃𝑛 ∈ ω 𝑥 = ((𝑖𝑔𝑗)⊼𝑔(𝑘𝑔𝑛)) ∨ 𝑥 = ∀𝑔𝑘(𝑖𝑔𝑗)) ↔ (∃𝑛 ∈ ω 𝑋 = ((𝑖𝑔𝑗)⊼𝑔(𝑘𝑔𝑛)) ∨ 𝑋 = ∀𝑔𝑘(𝑖𝑔𝑗))))
4544rexbidv 3175 . . . . . . 7 (𝑥 = 𝑋 → (∃𝑘 ∈ ω (∃𝑛 ∈ ω 𝑥 = ((𝑖𝑔𝑗)⊼𝑔(𝑘𝑔𝑛)) ∨ 𝑥 = ∀𝑔𝑘(𝑖𝑔𝑗)) ↔ ∃𝑘 ∈ ω (∃𝑛 ∈ ω 𝑋 = ((𝑖𝑔𝑗)⊼𝑔(𝑘𝑔𝑛)) ∨ 𝑋 = ∀𝑔𝑘(𝑖𝑔𝑗))))
46452rexbidv 3213 . . . . . 6 (𝑥 = 𝑋 → (∃𝑖 ∈ ω ∃𝑗 ∈ ω ∃𝑘 ∈ ω (∃𝑛 ∈ ω 𝑥 = ((𝑖𝑔𝑗)⊼𝑔(𝑘𝑔𝑛)) ∨ 𝑥 = ∀𝑔𝑘(𝑖𝑔𝑗)) ↔ ∃𝑖 ∈ ω ∃𝑗 ∈ ω ∃𝑘 ∈ ω (∃𝑛 ∈ ω 𝑋 = ((𝑖𝑔𝑗)⊼𝑔(𝑘𝑔𝑛)) ∨ 𝑋 = ∀𝑔𝑘(𝑖𝑔𝑗))))
4740, 46elab 3630 . . . . 5 (𝑋 ∈ {𝑥 ∣ ∃𝑖 ∈ ω ∃𝑗 ∈ ω ∃𝑘 ∈ ω (∃𝑛 ∈ ω 𝑥 = ((𝑖𝑔𝑗)⊼𝑔(𝑘𝑔𝑛)) ∨ 𝑥 = ∀𝑔𝑘(𝑖𝑔𝑗))} ↔ ∃𝑖 ∈ ω ∃𝑗 ∈ ω ∃𝑘 ∈ ω (∃𝑛 ∈ ω 𝑋 = ((𝑖𝑔𝑗)⊼𝑔(𝑘𝑔𝑛)) ∨ 𝑋 = ∀𝑔𝑘(𝑖𝑔𝑗)))
4839, 47sylibr 233 . . . 4 (((𝐼 ∈ ω ∧ 𝐽 ∈ ω) ∧ (𝐾 ∈ ω ∧ 𝐿 ∈ ω)) → 𝑋 ∈ {𝑥 ∣ ∃𝑖 ∈ ω ∃𝑗 ∈ ω ∃𝑘 ∈ ω (∃𝑛 ∈ ω 𝑥 = ((𝑖𝑔𝑗)⊼𝑔(𝑘𝑔𝑛)) ∨ 𝑥 = ∀𝑔𝑘(𝑖𝑔𝑗))})
4948olcd 872 . . 3 (((𝐼 ∈ ω ∧ 𝐽 ∈ ω) ∧ (𝐾 ∈ ω ∧ 𝐿 ∈ ω)) → (𝑋 ∈ ({∅} × (ω × ω)) ∨ 𝑋 ∈ {𝑥 ∣ ∃𝑖 ∈ ω ∃𝑗 ∈ ω ∃𝑘 ∈ ω (∃𝑛 ∈ ω 𝑥 = ((𝑖𝑔𝑗)⊼𝑔(𝑘𝑔𝑛)) ∨ 𝑥 = ∀𝑔𝑘(𝑖𝑔𝑗))}))
50 elun 4108 . . 3 (𝑋 ∈ (({∅} × (ω × ω)) ∪ {𝑥 ∣ ∃𝑖 ∈ ω ∃𝑗 ∈ ω ∃𝑘 ∈ ω (∃𝑛 ∈ ω 𝑥 = ((𝑖𝑔𝑗)⊼𝑔(𝑘𝑔𝑛)) ∨ 𝑥 = ∀𝑔𝑘(𝑖𝑔𝑗))}) ↔ (𝑋 ∈ ({∅} × (ω × ω)) ∨ 𝑋 ∈ {𝑥 ∣ ∃𝑖 ∈ ω ∃𝑗 ∈ ω ∃𝑘 ∈ ω (∃𝑛 ∈ ω 𝑥 = ((𝑖𝑔𝑗)⊼𝑔(𝑘𝑔𝑛)) ∨ 𝑥 = ∀𝑔𝑘(𝑖𝑔𝑗))}))
5149, 50sylibr 233 . 2 (((𝐼 ∈ ω ∧ 𝐽 ∈ ω) ∧ (𝐾 ∈ ω ∧ 𝐿 ∈ ω)) → 𝑋 ∈ (({∅} × (ω × ω)) ∪ {𝑥 ∣ ∃𝑖 ∈ ω ∃𝑗 ∈ ω ∃𝑘 ∈ ω (∃𝑛 ∈ ω 𝑥 = ((𝑖𝑔𝑗)⊼𝑔(𝑘𝑔𝑛)) ∨ 𝑥 = ∀𝑔𝑘(𝑖𝑔𝑗))}))
52 fmla1 33981 . 2 (Fmla‘1o) = (({∅} × (ω × ω)) ∪ {𝑥 ∣ ∃𝑖 ∈ ω ∃𝑗 ∈ ω ∃𝑘 ∈ ω (∃𝑛 ∈ ω 𝑥 = ((𝑖𝑔𝑗)⊼𝑔(𝑘𝑔𝑛)) ∨ 𝑥 = ∀𝑔𝑘(𝑖𝑔𝑗))})
5351, 52eleqtrrdi 2849 1 (((𝐼 ∈ ω ∧ 𝐽 ∈ ω) ∧ (𝐾 ∈ ω ∧ 𝐿 ∈ ω)) → 𝑋 ∈ (Fmla‘1o))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 396  wo 845   = wceq 1541  wcel 2106  {cab 2713  wrex 3073  cun 3908  c0 4282  {csn 4586   × cxp 5631  cfv 6496  (class class class)co 7357  ωcom 7802  1oc1o 8405  𝑔cgoe 33927  𝑔cgna 33928  𝑔cgol 33929  Fmlacfmla 33931
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2707  ax-rep 5242  ax-sep 5256  ax-nul 5263  ax-pow 5320  ax-pr 5384  ax-un 7672  ax-inf2 9577
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2889  df-ne 2944  df-nel 3050  df-ral 3065  df-rex 3074  df-reu 3354  df-rab 3408  df-v 3447  df-sbc 3740  df-csb 3856  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-pss 3929  df-nul 4283  df-if 4487  df-pw 4562  df-sn 4587  df-pr 4589  df-op 4593  df-uni 4866  df-iun 4956  df-br 5106  df-opab 5168  df-mpt 5189  df-tr 5223  df-id 5531  df-eprel 5537  df-po 5545  df-so 5546  df-fr 5588  df-we 5590  df-xp 5639  df-rel 5640  df-cnv 5641  df-co 5642  df-dm 5643  df-rn 5644  df-res 5645  df-ima 5646  df-pred 6253  df-ord 6320  df-on 6321  df-lim 6322  df-suc 6323  df-iota 6448  df-fun 6498  df-fn 6499  df-f 6500  df-f1 6501  df-fo 6502  df-f1o 6503  df-fv 6504  df-ov 7360  df-oprab 7361  df-mpo 7362  df-om 7803  df-1st 7921  df-2nd 7922  df-frecs 8212  df-wrecs 8243  df-recs 8317  df-rdg 8356  df-1o 8412  df-map 8767  df-goel 33934  df-goal 33936  df-sat 33937  df-fmla 33939
This theorem is referenced by:  satefvfmla1  34019
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