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Theorem 2goelgoanfmla1 35409
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 767 . . . . . 6 (((𝐼 ∈ ω ∧ 𝐽 ∈ ω) ∧ (𝐾 ∈ ω ∧ 𝐿 ∈ ω)) → 𝐼 ∈ ω)
2 simplr 769 . . . . . 6 (((𝐼 ∈ ω ∧ 𝐽 ∈ ω) ∧ (𝐾 ∈ ω ∧ 𝐿 ∈ ω)) → 𝐽 ∈ ω)
3 simprl 771 . . . . . 6 (((𝐼 ∈ ω ∧ 𝐽 ∈ ω) ∧ (𝐾 ∈ ω ∧ 𝐿 ∈ ω)) → 𝐾 ∈ ω)
4 simprr 773 . . . . . . . 8 (((𝐼 ∈ ω ∧ 𝐽 ∈ ω) ∧ (𝐾 ∈ ω ∧ 𝐿 ∈ ω)) → 𝐿 ∈ ω)
5 oveq2 7439 . . . . . . . . . . 11 (𝑛 = 𝐿 → (𝐾𝑔𝑛) = (𝐾𝑔𝐿))
65oveq2d 7447 . . . . . . . . . 10 (𝑛 = 𝐿 → ((𝐼𝑔𝐽)⊼𝑔(𝐾𝑔𝑛)) = ((𝐼𝑔𝐽)⊼𝑔(𝐾𝑔𝐿)))
76eqeq2d 2746 . . . . . . . . 9 (𝑛 = 𝐿 → (𝑋 = ((𝐼𝑔𝐽)⊼𝑔(𝐾𝑔𝑛)) ↔ 𝑋 = ((𝐼𝑔𝐽)⊼𝑔(𝐾𝑔𝐿))))
87adantl 481 . . . . . . . 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 3624 . . . . . . 7 (((𝐼 ∈ ω ∧ 𝐽 ∈ ω) ∧ (𝐾 ∈ ω ∧ 𝐿 ∈ ω)) → ∃𝑛 ∈ ω 𝑋 = ((𝐼𝑔𝐽)⊼𝑔(𝐾𝑔𝑛)))
1211orcd 873 . . . . . 6 (((𝐼 ∈ ω ∧ 𝐽 ∈ ω) ∧ (𝐾 ∈ ω ∧ 𝐿 ∈ ω)) → (∃𝑛 ∈ ω 𝑋 = ((𝐼𝑔𝐽)⊼𝑔(𝐾𝑔𝑛)) ∨ 𝑋 = ∀𝑔𝐾(𝐼𝑔𝐽)))
13 oveq1 7438 . . . . . . . . . . 11 (𝑖 = 𝐼 → (𝑖𝑔𝑗) = (𝐼𝑔𝑗))
1413oveq1d 7446 . . . . . . . . . 10 (𝑖 = 𝐼 → ((𝑖𝑔𝑗)⊼𝑔(𝑘𝑔𝑛)) = ((𝐼𝑔𝑗)⊼𝑔(𝑘𝑔𝑛)))
1514eqeq2d 2746 . . . . . . . . 9 (𝑖 = 𝐼 → (𝑋 = ((𝑖𝑔𝑗)⊼𝑔(𝑘𝑔𝑛)) ↔ 𝑋 = ((𝐼𝑔𝑗)⊼𝑔(𝑘𝑔𝑛))))
1615rexbidv 3177 . . . . . . . 8 (𝑖 = 𝐼 → (∃𝑛 ∈ ω 𝑋 = ((𝑖𝑔𝑗)⊼𝑔(𝑘𝑔𝑛)) ↔ ∃𝑛 ∈ ω 𝑋 = ((𝐼𝑔𝑗)⊼𝑔(𝑘𝑔𝑛))))
17 eqidd 2736 . . . . . . . . . 10 (𝑖 = 𝐼𝑘 = 𝑘)
1817, 13goaleq12d 35336 . . . . . . . . 9 (𝑖 = 𝐼 → ∀𝑔𝑘(𝑖𝑔𝑗) = ∀𝑔𝑘(𝐼𝑔𝑗))
1918eqeq2d 2746 . . . . . . . 8 (𝑖 = 𝐼 → (𝑋 = ∀𝑔𝑘(𝑖𝑔𝑗) ↔ 𝑋 = ∀𝑔𝑘(𝐼𝑔𝑗)))
2016, 19orbi12d 918 . . . . . . 7 (𝑖 = 𝐼 → ((∃𝑛 ∈ ω 𝑋 = ((𝑖𝑔𝑗)⊼𝑔(𝑘𝑔𝑛)) ∨ 𝑋 = ∀𝑔𝑘(𝑖𝑔𝑗)) ↔ (∃𝑛 ∈ ω 𝑋 = ((𝐼𝑔𝑗)⊼𝑔(𝑘𝑔𝑛)) ∨ 𝑋 = ∀𝑔𝑘(𝐼𝑔𝑗))))
21 oveq2 7439 . . . . . . . . . . 11 (𝑗 = 𝐽 → (𝐼𝑔𝑗) = (𝐼𝑔𝐽))
2221oveq1d 7446 . . . . . . . . . 10 (𝑗 = 𝐽 → ((𝐼𝑔𝑗)⊼𝑔(𝑘𝑔𝑛)) = ((𝐼𝑔𝐽)⊼𝑔(𝑘𝑔𝑛)))
2322eqeq2d 2746 . . . . . . . . 9 (𝑗 = 𝐽 → (𝑋 = ((𝐼𝑔𝑗)⊼𝑔(𝑘𝑔𝑛)) ↔ 𝑋 = ((𝐼𝑔𝐽)⊼𝑔(𝑘𝑔𝑛))))
2423rexbidv 3177 . . . . . . . 8 (𝑗 = 𝐽 → (∃𝑛 ∈ ω 𝑋 = ((𝐼𝑔𝑗)⊼𝑔(𝑘𝑔𝑛)) ↔ ∃𝑛 ∈ ω 𝑋 = ((𝐼𝑔𝐽)⊼𝑔(𝑘𝑔𝑛))))
25 eqidd 2736 . . . . . . . . . 10 (𝑗 = 𝐽𝑘 = 𝑘)
2625, 21goaleq12d 35336 . . . . . . . . 9 (𝑗 = 𝐽 → ∀𝑔𝑘(𝐼𝑔𝑗) = ∀𝑔𝑘(𝐼𝑔𝐽))
2726eqeq2d 2746 . . . . . . . 8 (𝑗 = 𝐽 → (𝑋 = ∀𝑔𝑘(𝐼𝑔𝑗) ↔ 𝑋 = ∀𝑔𝑘(𝐼𝑔𝐽)))
2824, 27orbi12d 918 . . . . . . 7 (𝑗 = 𝐽 → ((∃𝑛 ∈ ω 𝑋 = ((𝐼𝑔𝑗)⊼𝑔(𝑘𝑔𝑛)) ∨ 𝑋 = ∀𝑔𝑘(𝐼𝑔𝑗)) ↔ (∃𝑛 ∈ ω 𝑋 = ((𝐼𝑔𝐽)⊼𝑔(𝑘𝑔𝑛)) ∨ 𝑋 = ∀𝑔𝑘(𝐼𝑔𝐽))))
29 oveq1 7438 . . . . . . . . . . 11 (𝑘 = 𝐾 → (𝑘𝑔𝑛) = (𝐾𝑔𝑛))
3029oveq2d 7447 . . . . . . . . . 10 (𝑘 = 𝐾 → ((𝐼𝑔𝐽)⊼𝑔(𝑘𝑔𝑛)) = ((𝐼𝑔𝐽)⊼𝑔(𝐾𝑔𝑛)))
3130eqeq2d 2746 . . . . . . . . 9 (𝑘 = 𝐾 → (𝑋 = ((𝐼𝑔𝐽)⊼𝑔(𝑘𝑔𝑛)) ↔ 𝑋 = ((𝐼𝑔𝐽)⊼𝑔(𝐾𝑔𝑛))))
3231rexbidv 3177 . . . . . . . 8 (𝑘 = 𝐾 → (∃𝑛 ∈ ω 𝑋 = ((𝐼𝑔𝐽)⊼𝑔(𝑘𝑔𝑛)) ↔ ∃𝑛 ∈ ω 𝑋 = ((𝐼𝑔𝐽)⊼𝑔(𝐾𝑔𝑛))))
33 id 22 . . . . . . . . . 10 (𝑘 = 𝐾𝑘 = 𝐾)
34 eqidd 2736 . . . . . . . . . 10 (𝑘 = 𝐾 → (𝐼𝑔𝐽) = (𝐼𝑔𝐽))
3533, 34goaleq12d 35336 . . . . . . . . 9 (𝑘 = 𝐾 → ∀𝑔𝑘(𝐼𝑔𝐽) = ∀𝑔𝐾(𝐼𝑔𝐽))
3635eqeq2d 2746 . . . . . . . 8 (𝑘 = 𝐾 → (𝑋 = ∀𝑔𝑘(𝐼𝑔𝐽) ↔ 𝑋 = ∀𝑔𝐾(𝐼𝑔𝐽)))
3732, 36orbi12d 918 . . . . . . 7 (𝑘 = 𝐾 → ((∃𝑛 ∈ ω 𝑋 = ((𝐼𝑔𝐽)⊼𝑔(𝑘𝑔𝑛)) ∨ 𝑋 = ∀𝑔𝑘(𝐼𝑔𝐽)) ↔ (∃𝑛 ∈ ω 𝑋 = ((𝐼𝑔𝐽)⊼𝑔(𝐾𝑔𝑛)) ∨ 𝑋 = ∀𝑔𝐾(𝐼𝑔𝐽))))
3820, 28, 37rspc3ev 3639 . . . . . 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 1372 . . . . 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 7465 . . . . . 6 𝑋 ∈ V
41 eqeq1 2739 . . . . . . . . . 10 (𝑥 = 𝑋 → (𝑥 = ((𝑖𝑔𝑗)⊼𝑔(𝑘𝑔𝑛)) ↔ 𝑋 = ((𝑖𝑔𝑗)⊼𝑔(𝑘𝑔𝑛))))
4241rexbidv 3177 . . . . . . . . 9 (𝑥 = 𝑋 → (∃𝑛 ∈ ω 𝑥 = ((𝑖𝑔𝑗)⊼𝑔(𝑘𝑔𝑛)) ↔ ∃𝑛 ∈ ω 𝑋 = ((𝑖𝑔𝑗)⊼𝑔(𝑘𝑔𝑛))))
43 eqeq1 2739 . . . . . . . . 9 (𝑥 = 𝑋 → (𝑥 = ∀𝑔𝑘(𝑖𝑔𝑗) ↔ 𝑋 = ∀𝑔𝑘(𝑖𝑔𝑗)))
4442, 43orbi12d 918 . . . . . . . 8 (𝑥 = 𝑋 → ((∃𝑛 ∈ ω 𝑥 = ((𝑖𝑔𝑗)⊼𝑔(𝑘𝑔𝑛)) ∨ 𝑥 = ∀𝑔𝑘(𝑖𝑔𝑗)) ↔ (∃𝑛 ∈ ω 𝑋 = ((𝑖𝑔𝑗)⊼𝑔(𝑘𝑔𝑛)) ∨ 𝑋 = ∀𝑔𝑘(𝑖𝑔𝑗))))
4544rexbidv 3177 . . . . . . 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 3220 . . . . . 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 3681 . . . . 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 234 . . . 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 874 . . 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 4163 . . 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 234 . 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 35372 . 2 (Fmla‘1o) = (({∅} × (ω × ω)) ∪ {𝑥 ∣ ∃𝑖 ∈ ω ∃𝑗 ∈ ω ∃𝑘 ∈ ω (∃𝑛 ∈ ω 𝑥 = ((𝑖𝑔𝑗)⊼𝑔(𝑘𝑔𝑛)) ∨ 𝑥 = ∀𝑔𝑘(𝑖𝑔𝑗))})
5351, 52eleqtrrdi 2850 1 (((𝐼 ∈ ω ∧ 𝐽 ∈ ω) ∧ (𝐾 ∈ ω ∧ 𝐿 ∈ ω)) → 𝑋 ∈ (Fmla‘1o))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395  wo 847   = wceq 1537  wcel 2106  {cab 2712  wrex 3068  cun 3961  c0 4339  {csn 4631   × cxp 5687  cfv 6563  (class class class)co 7431  ωcom 7887  1oc1o 8498  𝑔cgoe 35318  𝑔cgna 35319  𝑔cgol 35320  Fmlacfmla 35322
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-rep 5285  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754  ax-inf2 9679
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  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-ne 2939  df-nel 3045  df-ral 3060  df-rex 3069  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-pss 3983  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5583  df-eprel 5589  df-po 5597  df-so 5598  df-fr 5641  df-we 5643  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-pred 6323  df-ord 6389  df-on 6390  df-lim 6391  df-suc 6392  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-ov 7434  df-oprab 7435  df-mpo 7436  df-om 7888  df-1st 8013  df-2nd 8014  df-frecs 8305  df-wrecs 8336  df-recs 8410  df-rdg 8449  df-1o 8505  df-map 8867  df-goel 35325  df-goal 35327  df-sat 35328  df-fmla 35330
This theorem is referenced by:  satefvfmla1  35410
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