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Theorem 2goelgoanfmla1 35429
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 2748 . . . . . . . . 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 874 . . . . . 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 2748 . . . . . . . . 9 (𝑖 = 𝐼 → (𝑋 = ((𝑖𝑔𝑗)⊼𝑔(𝑘𝑔𝑛)) ↔ 𝑋 = ((𝐼𝑔𝑗)⊼𝑔(𝑘𝑔𝑛))))
1615rexbidv 3179 . . . . . . . 8 (𝑖 = 𝐼 → (∃𝑛 ∈ ω 𝑋 = ((𝑖𝑔𝑗)⊼𝑔(𝑘𝑔𝑛)) ↔ ∃𝑛 ∈ ω 𝑋 = ((𝐼𝑔𝑗)⊼𝑔(𝑘𝑔𝑛))))
17 eqidd 2738 . . . . . . . . . 10 (𝑖 = 𝐼𝑘 = 𝑘)
1817, 13goaleq12d 35356 . . . . . . . . 9 (𝑖 = 𝐼 → ∀𝑔𝑘(𝑖𝑔𝑗) = ∀𝑔𝑘(𝐼𝑔𝑗))
1918eqeq2d 2748 . . . . . . . 8 (𝑖 = 𝐼 → (𝑋 = ∀𝑔𝑘(𝑖𝑔𝑗) ↔ 𝑋 = ∀𝑔𝑘(𝐼𝑔𝑗)))
2016, 19orbi12d 919 . . . . . . 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 2748 . . . . . . . . 9 (𝑗 = 𝐽 → (𝑋 = ((𝐼𝑔𝑗)⊼𝑔(𝑘𝑔𝑛)) ↔ 𝑋 = ((𝐼𝑔𝐽)⊼𝑔(𝑘𝑔𝑛))))
2423rexbidv 3179 . . . . . . . 8 (𝑗 = 𝐽 → (∃𝑛 ∈ ω 𝑋 = ((𝐼𝑔𝑗)⊼𝑔(𝑘𝑔𝑛)) ↔ ∃𝑛 ∈ ω 𝑋 = ((𝐼𝑔𝐽)⊼𝑔(𝑘𝑔𝑛))))
25 eqidd 2738 . . . . . . . . . 10 (𝑗 = 𝐽𝑘 = 𝑘)
2625, 21goaleq12d 35356 . . . . . . . . 9 (𝑗 = 𝐽 → ∀𝑔𝑘(𝐼𝑔𝑗) = ∀𝑔𝑘(𝐼𝑔𝐽))
2726eqeq2d 2748 . . . . . . . 8 (𝑗 = 𝐽 → (𝑋 = ∀𝑔𝑘(𝐼𝑔𝑗) ↔ 𝑋 = ∀𝑔𝑘(𝐼𝑔𝐽)))
2824, 27orbi12d 919 . . . . . . 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 2748 . . . . . . . . 9 (𝑘 = 𝐾 → (𝑋 = ((𝐼𝑔𝐽)⊼𝑔(𝑘𝑔𝑛)) ↔ 𝑋 = ((𝐼𝑔𝐽)⊼𝑔(𝐾𝑔𝑛))))
3231rexbidv 3179 . . . . . . . 8 (𝑘 = 𝐾 → (∃𝑛 ∈ ω 𝑋 = ((𝐼𝑔𝐽)⊼𝑔(𝑘𝑔𝑛)) ↔ ∃𝑛 ∈ ω 𝑋 = ((𝐼𝑔𝐽)⊼𝑔(𝐾𝑔𝑛))))
33 id 22 . . . . . . . . . 10 (𝑘 = 𝐾𝑘 = 𝐾)
34 eqidd 2738 . . . . . . . . . 10 (𝑘 = 𝐾 → (𝐼𝑔𝐽) = (𝐼𝑔𝐽))
3533, 34goaleq12d 35356 . . . . . . . . 9 (𝑘 = 𝐾 → ∀𝑔𝑘(𝐼𝑔𝐽) = ∀𝑔𝐾(𝐼𝑔𝐽))
3635eqeq2d 2748 . . . . . . . 8 (𝑘 = 𝐾 → (𝑋 = ∀𝑔𝑘(𝐼𝑔𝐽) ↔ 𝑋 = ∀𝑔𝐾(𝐼𝑔𝐽)))
3732, 36orbi12d 919 . . . . . . 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 1375 . . . . 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 2741 . . . . . . . . . 10 (𝑥 = 𝑋 → (𝑥 = ((𝑖𝑔𝑗)⊼𝑔(𝑘𝑔𝑛)) ↔ 𝑋 = ((𝑖𝑔𝑗)⊼𝑔(𝑘𝑔𝑛))))
4241rexbidv 3179 . . . . . . . . 9 (𝑥 = 𝑋 → (∃𝑛 ∈ ω 𝑥 = ((𝑖𝑔𝑗)⊼𝑔(𝑘𝑔𝑛)) ↔ ∃𝑛 ∈ ω 𝑋 = ((𝑖𝑔𝑗)⊼𝑔(𝑘𝑔𝑛))))
43 eqeq1 2741 . . . . . . . . 9 (𝑥 = 𝑋 → (𝑥 = ∀𝑔𝑘(𝑖𝑔𝑗) ↔ 𝑋 = ∀𝑔𝑘(𝑖𝑔𝑗)))
4442, 43orbi12d 919 . . . . . . . 8 (𝑥 = 𝑋 → ((∃𝑛 ∈ ω 𝑥 = ((𝑖𝑔𝑗)⊼𝑔(𝑘𝑔𝑛)) ∨ 𝑥 = ∀𝑔𝑘(𝑖𝑔𝑗)) ↔ (∃𝑛 ∈ ω 𝑋 = ((𝑖𝑔𝑗)⊼𝑔(𝑘𝑔𝑛)) ∨ 𝑋 = ∀𝑔𝑘(𝑖𝑔𝑗))))
4544rexbidv 3179 . . . . . . 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 3222 . . . . . 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 3679 . . . . 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 875 . . 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 4153 . . 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 35392 . 2 (Fmla‘1o) = (({∅} × (ω × ω)) ∪ {𝑥 ∣ ∃𝑖 ∈ ω ∃𝑗 ∈ ω ∃𝑘 ∈ ω (∃𝑛 ∈ ω 𝑥 = ((𝑖𝑔𝑗)⊼𝑔(𝑘𝑔𝑛)) ∨ 𝑥 = ∀𝑔𝑘(𝑖𝑔𝑗))})
5351, 52eleqtrrdi 2852 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 848   = wceq 1540  wcel 2108  {cab 2714  wrex 3070  cun 3949  c0 4333  {csn 4626   × cxp 5683  cfv 6561  (class class class)co 7431  ωcom 7887  1oc1o 8499  𝑔cgoe 35338  𝑔cgna 35339  𝑔cgol 35340  Fmlacfmla 35342
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 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-rep 5279  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755  ax-inf2 9681
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-pss 3971  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-tr 5260  df-id 5578  df-eprel 5584  df-po 5592  df-so 5593  df-fr 5637  df-we 5639  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-pred 6321  df-ord 6387  df-on 6388  df-lim 6389  df-suc 6390  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-ov 7434  df-oprab 7435  df-mpo 7436  df-om 7888  df-1st 8014  df-2nd 8015  df-frecs 8306  df-wrecs 8337  df-recs 8411  df-rdg 8450  df-1o 8506  df-map 8868  df-goel 35345  df-goal 35347  df-sat 35348  df-fmla 35350
This theorem is referenced by:  satefvfmla1  35430
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