Metamath Proof Explorer


Theorem fmla1

Description: The valid Godel formulas of height 1 is the set of all formulas of the form ( a |g b ) and A.g k a with atoms a , b of the form x e. y . (Contributed by AV, 20-Sep-2023)

Ref Expression
Assertion fmla1 ( Fmla ‘ 1o ) = ( ( { ∅ } × ( ω × ω ) ) ∪ { 𝑥 ∣ ∃ 𝑖 ∈ ω ∃ 𝑗 ∈ ω ∃ 𝑘 ∈ ω ( ∃ 𝑙 ∈ ω 𝑥 = ( ( 𝑖𝑔 𝑗 ) ⊼𝑔 ( 𝑘𝑔 𝑙 ) ) ∨ 𝑥 = ∀𝑔 𝑘 ( 𝑖𝑔 𝑗 ) ) } )

Proof

Step Hyp Ref Expression
1 df-1o 1o = suc ∅
2 1 fveq2i ( Fmla ‘ 1o ) = ( Fmla ‘ suc ∅ )
3 peano1 ∅ ∈ ω
4 fmlasuc ( ∅ ∈ ω → ( Fmla ‘ suc ∅ ) = ( ( Fmla ‘ ∅ ) ∪ { 𝑥 ∣ ∃ 𝑝 ∈ ( Fmla ‘ ∅ ) ( ∃ 𝑞 ∈ ( Fmla ‘ ∅ ) 𝑥 = ( 𝑝𝑔 𝑞 ) ∨ ∃ 𝑘 ∈ ω 𝑥 = ∀𝑔 𝑘 𝑝 ) } ) )
5 3 4 ax-mp ( Fmla ‘ suc ∅ ) = ( ( Fmla ‘ ∅ ) ∪ { 𝑥 ∣ ∃ 𝑝 ∈ ( Fmla ‘ ∅ ) ( ∃ 𝑞 ∈ ( Fmla ‘ ∅ ) 𝑥 = ( 𝑝𝑔 𝑞 ) ∨ ∃ 𝑘 ∈ ω 𝑥 = ∀𝑔 𝑘 𝑝 ) } )
6 fmla0xp ( Fmla ‘ ∅ ) = ( { ∅ } × ( ω × ω ) )
7 fmla0 ( Fmla ‘ ∅ ) = { 𝑦 ∈ V ∣ ∃ 𝑖 ∈ ω ∃ 𝑗 ∈ ω 𝑦 = ( 𝑖𝑔 𝑗 ) }
8 7 rexeqi ( ∃ 𝑝 ∈ ( Fmla ‘ ∅ ) ( ∃ 𝑞 ∈ ( Fmla ‘ ∅ ) 𝑥 = ( 𝑝𝑔 𝑞 ) ∨ ∃ 𝑘 ∈ ω 𝑥 = ∀𝑔 𝑘 𝑝 ) ↔ ∃ 𝑝 ∈ { 𝑦 ∈ V ∣ ∃ 𝑖 ∈ ω ∃ 𝑗 ∈ ω 𝑦 = ( 𝑖𝑔 𝑗 ) } ( ∃ 𝑞 ∈ ( Fmla ‘ ∅ ) 𝑥 = ( 𝑝𝑔 𝑞 ) ∨ ∃ 𝑘 ∈ ω 𝑥 = ∀𝑔 𝑘 𝑝 ) )
9 eqeq1 ( 𝑦 = 𝑝 → ( 𝑦 = ( 𝑖𝑔 𝑗 ) ↔ 𝑝 = ( 𝑖𝑔 𝑗 ) ) )
10 9 2rexbidv ( 𝑦 = 𝑝 → ( ∃ 𝑖 ∈ ω ∃ 𝑗 ∈ ω 𝑦 = ( 𝑖𝑔 𝑗 ) ↔ ∃ 𝑖 ∈ ω ∃ 𝑗 ∈ ω 𝑝 = ( 𝑖𝑔 𝑗 ) ) )
11 10 elrab ( 𝑝 ∈ { 𝑦 ∈ V ∣ ∃ 𝑖 ∈ ω ∃ 𝑗 ∈ ω 𝑦 = ( 𝑖𝑔 𝑗 ) } ↔ ( 𝑝 ∈ V ∧ ∃ 𝑖 ∈ ω ∃ 𝑗 ∈ ω 𝑝 = ( 𝑖𝑔 𝑗 ) ) )
12 oveq1 ( 𝑝 = ( 𝑖𝑔 𝑗 ) → ( 𝑝𝑔 𝑞 ) = ( ( 𝑖𝑔 𝑗 ) ⊼𝑔 𝑞 ) )
13 12 eqeq2d ( 𝑝 = ( 𝑖𝑔 𝑗 ) → ( 𝑥 = ( 𝑝𝑔 𝑞 ) ↔ 𝑥 = ( ( 𝑖𝑔 𝑗 ) ⊼𝑔 𝑞 ) ) )
14 13 rexbidv ( 𝑝 = ( 𝑖𝑔 𝑗 ) → ( ∃ 𝑞 ∈ ( Fmla ‘ ∅ ) 𝑥 = ( 𝑝𝑔 𝑞 ) ↔ ∃ 𝑞 ∈ ( Fmla ‘ ∅ ) 𝑥 = ( ( 𝑖𝑔 𝑗 ) ⊼𝑔 𝑞 ) ) )
15 eqidd ( 𝑝 = ( 𝑖𝑔 𝑗 ) → 𝑘 = 𝑘 )
16 id ( 𝑝 = ( 𝑖𝑔 𝑗 ) → 𝑝 = ( 𝑖𝑔 𝑗 ) )
17 15 16 goaleq12d ( 𝑝 = ( 𝑖𝑔 𝑗 ) → ∀𝑔 𝑘 𝑝 = ∀𝑔 𝑘 ( 𝑖𝑔 𝑗 ) )
18 17 eqeq2d ( 𝑝 = ( 𝑖𝑔 𝑗 ) → ( 𝑥 = ∀𝑔 𝑘 𝑝𝑥 = ∀𝑔 𝑘 ( 𝑖𝑔 𝑗 ) ) )
19 18 rexbidv ( 𝑝 = ( 𝑖𝑔 𝑗 ) → ( ∃ 𝑘 ∈ ω 𝑥 = ∀𝑔 𝑘 𝑝 ↔ ∃ 𝑘 ∈ ω 𝑥 = ∀𝑔 𝑘 ( 𝑖𝑔 𝑗 ) ) )
20 14 19 orbi12d ( 𝑝 = ( 𝑖𝑔 𝑗 ) → ( ( ∃ 𝑞 ∈ ( Fmla ‘ ∅ ) 𝑥 = ( 𝑝𝑔 𝑞 ) ∨ ∃ 𝑘 ∈ ω 𝑥 = ∀𝑔 𝑘 𝑝 ) ↔ ( ∃ 𝑞 ∈ ( Fmla ‘ ∅ ) 𝑥 = ( ( 𝑖𝑔 𝑗 ) ⊼𝑔 𝑞 ) ∨ ∃ 𝑘 ∈ ω 𝑥 = ∀𝑔 𝑘 ( 𝑖𝑔 𝑗 ) ) ) )
21 eqeq1 ( 𝑧 = 𝑞 → ( 𝑧 = ( 𝑘𝑔 𝑙 ) ↔ 𝑞 = ( 𝑘𝑔 𝑙 ) ) )
22 21 2rexbidv ( 𝑧 = 𝑞 → ( ∃ 𝑘 ∈ ω ∃ 𝑙 ∈ ω 𝑧 = ( 𝑘𝑔 𝑙 ) ↔ ∃ 𝑘 ∈ ω ∃ 𝑙 ∈ ω 𝑞 = ( 𝑘𝑔 𝑙 ) ) )
23 fmla0 ( Fmla ‘ ∅ ) = { 𝑧 ∈ V ∣ ∃ 𝑘 ∈ ω ∃ 𝑙 ∈ ω 𝑧 = ( 𝑘𝑔 𝑙 ) }
24 22 23 elrab2 ( 𝑞 ∈ ( Fmla ‘ ∅ ) ↔ ( 𝑞 ∈ V ∧ ∃ 𝑘 ∈ ω ∃ 𝑙 ∈ ω 𝑞 = ( 𝑘𝑔 𝑙 ) ) )
25 oveq2 ( 𝑞 = ( 𝑘𝑔 𝑙 ) → ( ( 𝑖𝑔 𝑗 ) ⊼𝑔 𝑞 ) = ( ( 𝑖𝑔 𝑗 ) ⊼𝑔 ( 𝑘𝑔 𝑙 ) ) )
26 25 eqeq2d ( 𝑞 = ( 𝑘𝑔 𝑙 ) → ( 𝑥 = ( ( 𝑖𝑔 𝑗 ) ⊼𝑔 𝑞 ) ↔ 𝑥 = ( ( 𝑖𝑔 𝑗 ) ⊼𝑔 ( 𝑘𝑔 𝑙 ) ) ) )
27 26 biimpcd ( 𝑥 = ( ( 𝑖𝑔 𝑗 ) ⊼𝑔 𝑞 ) → ( 𝑞 = ( 𝑘𝑔 𝑙 ) → 𝑥 = ( ( 𝑖𝑔 𝑗 ) ⊼𝑔 ( 𝑘𝑔 𝑙 ) ) ) )
28 27 reximdv ( 𝑥 = ( ( 𝑖𝑔 𝑗 ) ⊼𝑔 𝑞 ) → ( ∃ 𝑙 ∈ ω 𝑞 = ( 𝑘𝑔 𝑙 ) → ∃ 𝑙 ∈ ω 𝑥 = ( ( 𝑖𝑔 𝑗 ) ⊼𝑔 ( 𝑘𝑔 𝑙 ) ) ) )
29 28 reximdv ( 𝑥 = ( ( 𝑖𝑔 𝑗 ) ⊼𝑔 𝑞 ) → ( ∃ 𝑘 ∈ ω ∃ 𝑙 ∈ ω 𝑞 = ( 𝑘𝑔 𝑙 ) → ∃ 𝑘 ∈ ω ∃ 𝑙 ∈ ω 𝑥 = ( ( 𝑖𝑔 𝑗 ) ⊼𝑔 ( 𝑘𝑔 𝑙 ) ) ) )
30 29 com12 ( ∃ 𝑘 ∈ ω ∃ 𝑙 ∈ ω 𝑞 = ( 𝑘𝑔 𝑙 ) → ( 𝑥 = ( ( 𝑖𝑔 𝑗 ) ⊼𝑔 𝑞 ) → ∃ 𝑘 ∈ ω ∃ 𝑙 ∈ ω 𝑥 = ( ( 𝑖𝑔 𝑗 ) ⊼𝑔 ( 𝑘𝑔 𝑙 ) ) ) )
31 24 30 simplbiim ( 𝑞 ∈ ( Fmla ‘ ∅ ) → ( 𝑥 = ( ( 𝑖𝑔 𝑗 ) ⊼𝑔 𝑞 ) → ∃ 𝑘 ∈ ω ∃ 𝑙 ∈ ω 𝑥 = ( ( 𝑖𝑔 𝑗 ) ⊼𝑔 ( 𝑘𝑔 𝑙 ) ) ) )
32 31 rexlimiv ( ∃ 𝑞 ∈ ( Fmla ‘ ∅ ) 𝑥 = ( ( 𝑖𝑔 𝑗 ) ⊼𝑔 𝑞 ) → ∃ 𝑘 ∈ ω ∃ 𝑙 ∈ ω 𝑥 = ( ( 𝑖𝑔 𝑗 ) ⊼𝑔 ( 𝑘𝑔 𝑙 ) ) )
33 32 orim1i ( ( ∃ 𝑞 ∈ ( Fmla ‘ ∅ ) 𝑥 = ( ( 𝑖𝑔 𝑗 ) ⊼𝑔 𝑞 ) ∨ ∃ 𝑘 ∈ ω 𝑥 = ∀𝑔 𝑘 ( 𝑖𝑔 𝑗 ) ) → ( ∃ 𝑘 ∈ ω ∃ 𝑙 ∈ ω 𝑥 = ( ( 𝑖𝑔 𝑗 ) ⊼𝑔 ( 𝑘𝑔 𝑙 ) ) ∨ ∃ 𝑘 ∈ ω 𝑥 = ∀𝑔 𝑘 ( 𝑖𝑔 𝑗 ) ) )
34 r19.43 ( ∃ 𝑘 ∈ ω ( ∃ 𝑙 ∈ ω 𝑥 = ( ( 𝑖𝑔 𝑗 ) ⊼𝑔 ( 𝑘𝑔 𝑙 ) ) ∨ 𝑥 = ∀𝑔 𝑘 ( 𝑖𝑔 𝑗 ) ) ↔ ( ∃ 𝑘 ∈ ω ∃ 𝑙 ∈ ω 𝑥 = ( ( 𝑖𝑔 𝑗 ) ⊼𝑔 ( 𝑘𝑔 𝑙 ) ) ∨ ∃ 𝑘 ∈ ω 𝑥 = ∀𝑔 𝑘 ( 𝑖𝑔 𝑗 ) ) )
35 33 34 sylibr ( ( ∃ 𝑞 ∈ ( Fmla ‘ ∅ ) 𝑥 = ( ( 𝑖𝑔 𝑗 ) ⊼𝑔 𝑞 ) ∨ ∃ 𝑘 ∈ ω 𝑥 = ∀𝑔 𝑘 ( 𝑖𝑔 𝑗 ) ) → ∃ 𝑘 ∈ ω ( ∃ 𝑙 ∈ ω 𝑥 = ( ( 𝑖𝑔 𝑗 ) ⊼𝑔 ( 𝑘𝑔 𝑙 ) ) ∨ 𝑥 = ∀𝑔 𝑘 ( 𝑖𝑔 𝑗 ) ) )
36 20 35 syl6bi ( 𝑝 = ( 𝑖𝑔 𝑗 ) → ( ( ∃ 𝑞 ∈ ( Fmla ‘ ∅ ) 𝑥 = ( 𝑝𝑔 𝑞 ) ∨ ∃ 𝑘 ∈ ω 𝑥 = ∀𝑔 𝑘 𝑝 ) → ∃ 𝑘 ∈ ω ( ∃ 𝑙 ∈ ω 𝑥 = ( ( 𝑖𝑔 𝑗 ) ⊼𝑔 ( 𝑘𝑔 𝑙 ) ) ∨ 𝑥 = ∀𝑔 𝑘 ( 𝑖𝑔 𝑗 ) ) ) )
37 36 com12 ( ( ∃ 𝑞 ∈ ( Fmla ‘ ∅ ) 𝑥 = ( 𝑝𝑔 𝑞 ) ∨ ∃ 𝑘 ∈ ω 𝑥 = ∀𝑔 𝑘 𝑝 ) → ( 𝑝 = ( 𝑖𝑔 𝑗 ) → ∃ 𝑘 ∈ ω ( ∃ 𝑙 ∈ ω 𝑥 = ( ( 𝑖𝑔 𝑗 ) ⊼𝑔 ( 𝑘𝑔 𝑙 ) ) ∨ 𝑥 = ∀𝑔 𝑘 ( 𝑖𝑔 𝑗 ) ) ) )
38 37 reximdv ( ( ∃ 𝑞 ∈ ( Fmla ‘ ∅ ) 𝑥 = ( 𝑝𝑔 𝑞 ) ∨ ∃ 𝑘 ∈ ω 𝑥 = ∀𝑔 𝑘 𝑝 ) → ( ∃ 𝑗 ∈ ω 𝑝 = ( 𝑖𝑔 𝑗 ) → ∃ 𝑗 ∈ ω ∃ 𝑘 ∈ ω ( ∃ 𝑙 ∈ ω 𝑥 = ( ( 𝑖𝑔 𝑗 ) ⊼𝑔 ( 𝑘𝑔 𝑙 ) ) ∨ 𝑥 = ∀𝑔 𝑘 ( 𝑖𝑔 𝑗 ) ) ) )
39 38 reximdv ( ( ∃ 𝑞 ∈ ( Fmla ‘ ∅ ) 𝑥 = ( 𝑝𝑔 𝑞 ) ∨ ∃ 𝑘 ∈ ω 𝑥 = ∀𝑔 𝑘 𝑝 ) → ( ∃ 𝑖 ∈ ω ∃ 𝑗 ∈ ω 𝑝 = ( 𝑖𝑔 𝑗 ) → ∃ 𝑖 ∈ ω ∃ 𝑗 ∈ ω ∃ 𝑘 ∈ ω ( ∃ 𝑙 ∈ ω 𝑥 = ( ( 𝑖𝑔 𝑗 ) ⊼𝑔 ( 𝑘𝑔 𝑙 ) ) ∨ 𝑥 = ∀𝑔 𝑘 ( 𝑖𝑔 𝑗 ) ) ) )
40 39 com12 ( ∃ 𝑖 ∈ ω ∃ 𝑗 ∈ ω 𝑝 = ( 𝑖𝑔 𝑗 ) → ( ( ∃ 𝑞 ∈ ( Fmla ‘ ∅ ) 𝑥 = ( 𝑝𝑔 𝑞 ) ∨ ∃ 𝑘 ∈ ω 𝑥 = ∀𝑔 𝑘 𝑝 ) → ∃ 𝑖 ∈ ω ∃ 𝑗 ∈ ω ∃ 𝑘 ∈ ω ( ∃ 𝑙 ∈ ω 𝑥 = ( ( 𝑖𝑔 𝑗 ) ⊼𝑔 ( 𝑘𝑔 𝑙 ) ) ∨ 𝑥 = ∀𝑔 𝑘 ( 𝑖𝑔 𝑗 ) ) ) )
41 11 40 simplbiim ( 𝑝 ∈ { 𝑦 ∈ V ∣ ∃ 𝑖 ∈ ω ∃ 𝑗 ∈ ω 𝑦 = ( 𝑖𝑔 𝑗 ) } → ( ( ∃ 𝑞 ∈ ( Fmla ‘ ∅ ) 𝑥 = ( 𝑝𝑔 𝑞 ) ∨ ∃ 𝑘 ∈ ω 𝑥 = ∀𝑔 𝑘 𝑝 ) → ∃ 𝑖 ∈ ω ∃ 𝑗 ∈ ω ∃ 𝑘 ∈ ω ( ∃ 𝑙 ∈ ω 𝑥 = ( ( 𝑖𝑔 𝑗 ) ⊼𝑔 ( 𝑘𝑔 𝑙 ) ) ∨ 𝑥 = ∀𝑔 𝑘 ( 𝑖𝑔 𝑗 ) ) ) )
42 41 rexlimiv ( ∃ 𝑝 ∈ { 𝑦 ∈ V ∣ ∃ 𝑖 ∈ ω ∃ 𝑗 ∈ ω 𝑦 = ( 𝑖𝑔 𝑗 ) } ( ∃ 𝑞 ∈ ( Fmla ‘ ∅ ) 𝑥 = ( 𝑝𝑔 𝑞 ) ∨ ∃ 𝑘 ∈ ω 𝑥 = ∀𝑔 𝑘 𝑝 ) → ∃ 𝑖 ∈ ω ∃ 𝑗 ∈ ω ∃ 𝑘 ∈ ω ( ∃ 𝑙 ∈ ω 𝑥 = ( ( 𝑖𝑔 𝑗 ) ⊼𝑔 ( 𝑘𝑔 𝑙 ) ) ∨ 𝑥 = ∀𝑔 𝑘 ( 𝑖𝑔 𝑗 ) ) )
43 oveq1 ( 𝑖 = 𝑚 → ( 𝑖𝑔 𝑗 ) = ( 𝑚𝑔 𝑗 ) )
44 43 oveq1d ( 𝑖 = 𝑚 → ( ( 𝑖𝑔 𝑗 ) ⊼𝑔 ( 𝑘𝑔 𝑙 ) ) = ( ( 𝑚𝑔 𝑗 ) ⊼𝑔 ( 𝑘𝑔 𝑙 ) ) )
45 44 eqeq2d ( 𝑖 = 𝑚 → ( 𝑥 = ( ( 𝑖𝑔 𝑗 ) ⊼𝑔 ( 𝑘𝑔 𝑙 ) ) ↔ 𝑥 = ( ( 𝑚𝑔 𝑗 ) ⊼𝑔 ( 𝑘𝑔 𝑙 ) ) ) )
46 45 rexbidv ( 𝑖 = 𝑚 → ( ∃ 𝑙 ∈ ω 𝑥 = ( ( 𝑖𝑔 𝑗 ) ⊼𝑔 ( 𝑘𝑔 𝑙 ) ) ↔ ∃ 𝑙 ∈ ω 𝑥 = ( ( 𝑚𝑔 𝑗 ) ⊼𝑔 ( 𝑘𝑔 𝑙 ) ) ) )
47 eqidd ( 𝑖 = 𝑚𝑘 = 𝑘 )
48 47 43 goaleq12d ( 𝑖 = 𝑚 → ∀𝑔 𝑘 ( 𝑖𝑔 𝑗 ) = ∀𝑔 𝑘 ( 𝑚𝑔 𝑗 ) )
49 48 eqeq2d ( 𝑖 = 𝑚 → ( 𝑥 = ∀𝑔 𝑘 ( 𝑖𝑔 𝑗 ) ↔ 𝑥 = ∀𝑔 𝑘 ( 𝑚𝑔 𝑗 ) ) )
50 46 49 orbi12d ( 𝑖 = 𝑚 → ( ( ∃ 𝑙 ∈ ω 𝑥 = ( ( 𝑖𝑔 𝑗 ) ⊼𝑔 ( 𝑘𝑔 𝑙 ) ) ∨ 𝑥 = ∀𝑔 𝑘 ( 𝑖𝑔 𝑗 ) ) ↔ ( ∃ 𝑙 ∈ ω 𝑥 = ( ( 𝑚𝑔 𝑗 ) ⊼𝑔 ( 𝑘𝑔 𝑙 ) ) ∨ 𝑥 = ∀𝑔 𝑘 ( 𝑚𝑔 𝑗 ) ) ) )
51 50 rexbidv ( 𝑖 = 𝑚 → ( ∃ 𝑘 ∈ ω ( ∃ 𝑙 ∈ ω 𝑥 = ( ( 𝑖𝑔 𝑗 ) ⊼𝑔 ( 𝑘𝑔 𝑙 ) ) ∨ 𝑥 = ∀𝑔 𝑘 ( 𝑖𝑔 𝑗 ) ) ↔ ∃ 𝑘 ∈ ω ( ∃ 𝑙 ∈ ω 𝑥 = ( ( 𝑚𝑔 𝑗 ) ⊼𝑔 ( 𝑘𝑔 𝑙 ) ) ∨ 𝑥 = ∀𝑔 𝑘 ( 𝑚𝑔 𝑗 ) ) ) )
52 oveq2 ( 𝑗 = 𝑛 → ( 𝑚𝑔 𝑗 ) = ( 𝑚𝑔 𝑛 ) )
53 52 oveq1d ( 𝑗 = 𝑛 → ( ( 𝑚𝑔 𝑗 ) ⊼𝑔 ( 𝑘𝑔 𝑙 ) ) = ( ( 𝑚𝑔 𝑛 ) ⊼𝑔 ( 𝑘𝑔 𝑙 ) ) )
54 53 eqeq2d ( 𝑗 = 𝑛 → ( 𝑥 = ( ( 𝑚𝑔 𝑗 ) ⊼𝑔 ( 𝑘𝑔 𝑙 ) ) ↔ 𝑥 = ( ( 𝑚𝑔 𝑛 ) ⊼𝑔 ( 𝑘𝑔 𝑙 ) ) ) )
55 54 rexbidv ( 𝑗 = 𝑛 → ( ∃ 𝑙 ∈ ω 𝑥 = ( ( 𝑚𝑔 𝑗 ) ⊼𝑔 ( 𝑘𝑔 𝑙 ) ) ↔ ∃ 𝑙 ∈ ω 𝑥 = ( ( 𝑚𝑔 𝑛 ) ⊼𝑔 ( 𝑘𝑔 𝑙 ) ) ) )
56 eqidd ( 𝑗 = 𝑛𝑘 = 𝑘 )
57 56 52 goaleq12d ( 𝑗 = 𝑛 → ∀𝑔 𝑘 ( 𝑚𝑔 𝑗 ) = ∀𝑔 𝑘 ( 𝑚𝑔 𝑛 ) )
58 57 eqeq2d ( 𝑗 = 𝑛 → ( 𝑥 = ∀𝑔 𝑘 ( 𝑚𝑔 𝑗 ) ↔ 𝑥 = ∀𝑔 𝑘 ( 𝑚𝑔 𝑛 ) ) )
59 55 58 orbi12d ( 𝑗 = 𝑛 → ( ( ∃ 𝑙 ∈ ω 𝑥 = ( ( 𝑚𝑔 𝑗 ) ⊼𝑔 ( 𝑘𝑔 𝑙 ) ) ∨ 𝑥 = ∀𝑔 𝑘 ( 𝑚𝑔 𝑗 ) ) ↔ ( ∃ 𝑙 ∈ ω 𝑥 = ( ( 𝑚𝑔 𝑛 ) ⊼𝑔 ( 𝑘𝑔 𝑙 ) ) ∨ 𝑥 = ∀𝑔 𝑘 ( 𝑚𝑔 𝑛 ) ) ) )
60 59 rexbidv ( 𝑗 = 𝑛 → ( ∃ 𝑘 ∈ ω ( ∃ 𝑙 ∈ ω 𝑥 = ( ( 𝑚𝑔 𝑗 ) ⊼𝑔 ( 𝑘𝑔 𝑙 ) ) ∨ 𝑥 = ∀𝑔 𝑘 ( 𝑚𝑔 𝑗 ) ) ↔ ∃ 𝑘 ∈ ω ( ∃ 𝑙 ∈ ω 𝑥 = ( ( 𝑚𝑔 𝑛 ) ⊼𝑔 ( 𝑘𝑔 𝑙 ) ) ∨ 𝑥 = ∀𝑔 𝑘 ( 𝑚𝑔 𝑛 ) ) ) )
61 51 60 cbvrex2vw ( ∃ 𝑖 ∈ ω ∃ 𝑗 ∈ ω ∃ 𝑘 ∈ ω ( ∃ 𝑙 ∈ ω 𝑥 = ( ( 𝑖𝑔 𝑗 ) ⊼𝑔 ( 𝑘𝑔 𝑙 ) ) ∨ 𝑥 = ∀𝑔 𝑘 ( 𝑖𝑔 𝑗 ) ) ↔ ∃ 𝑚 ∈ ω ∃ 𝑛 ∈ ω ∃ 𝑘 ∈ ω ( ∃ 𝑙 ∈ ω 𝑥 = ( ( 𝑚𝑔 𝑛 ) ⊼𝑔 ( 𝑘𝑔 𝑙 ) ) ∨ 𝑥 = ∀𝑔 𝑘 ( 𝑚𝑔 𝑛 ) ) )
62 oveq1 ( 𝑘 = 𝑜 → ( 𝑘𝑔 𝑙 ) = ( 𝑜𝑔 𝑙 ) )
63 62 oveq2d ( 𝑘 = 𝑜 → ( ( 𝑚𝑔 𝑛 ) ⊼𝑔 ( 𝑘𝑔 𝑙 ) ) = ( ( 𝑚𝑔 𝑛 ) ⊼𝑔 ( 𝑜𝑔 𝑙 ) ) )
64 63 eqeq2d ( 𝑘 = 𝑜 → ( 𝑥 = ( ( 𝑚𝑔 𝑛 ) ⊼𝑔 ( 𝑘𝑔 𝑙 ) ) ↔ 𝑥 = ( ( 𝑚𝑔 𝑛 ) ⊼𝑔 ( 𝑜𝑔 𝑙 ) ) ) )
65 64 rexbidv ( 𝑘 = 𝑜 → ( ∃ 𝑙 ∈ ω 𝑥 = ( ( 𝑚𝑔 𝑛 ) ⊼𝑔 ( 𝑘𝑔 𝑙 ) ) ↔ ∃ 𝑙 ∈ ω 𝑥 = ( ( 𝑚𝑔 𝑛 ) ⊼𝑔 ( 𝑜𝑔 𝑙 ) ) ) )
66 id ( 𝑘 = 𝑜𝑘 = 𝑜 )
67 eqidd ( 𝑘 = 𝑜 → ( 𝑚𝑔 𝑛 ) = ( 𝑚𝑔 𝑛 ) )
68 66 67 goaleq12d ( 𝑘 = 𝑜 → ∀𝑔 𝑘 ( 𝑚𝑔 𝑛 ) = ∀𝑔 𝑜 ( 𝑚𝑔 𝑛 ) )
69 68 eqeq2d ( 𝑘 = 𝑜 → ( 𝑥 = ∀𝑔 𝑘 ( 𝑚𝑔 𝑛 ) ↔ 𝑥 = ∀𝑔 𝑜 ( 𝑚𝑔 𝑛 ) ) )
70 65 69 orbi12d ( 𝑘 = 𝑜 → ( ( ∃ 𝑙 ∈ ω 𝑥 = ( ( 𝑚𝑔 𝑛 ) ⊼𝑔 ( 𝑘𝑔 𝑙 ) ) ∨ 𝑥 = ∀𝑔 𝑘 ( 𝑚𝑔 𝑛 ) ) ↔ ( ∃ 𝑙 ∈ ω 𝑥 = ( ( 𝑚𝑔 𝑛 ) ⊼𝑔 ( 𝑜𝑔 𝑙 ) ) ∨ 𝑥 = ∀𝑔 𝑜 ( 𝑚𝑔 𝑛 ) ) ) )
71 70 cbvrexvw ( ∃ 𝑘 ∈ ω ( ∃ 𝑙 ∈ ω 𝑥 = ( ( 𝑚𝑔 𝑛 ) ⊼𝑔 ( 𝑘𝑔 𝑙 ) ) ∨ 𝑥 = ∀𝑔 𝑘 ( 𝑚𝑔 𝑛 ) ) ↔ ∃ 𝑜 ∈ ω ( ∃ 𝑙 ∈ ω 𝑥 = ( ( 𝑚𝑔 𝑛 ) ⊼𝑔 ( 𝑜𝑔 𝑙 ) ) ∨ 𝑥 = ∀𝑔 𝑜 ( 𝑚𝑔 𝑛 ) ) )
72 3 ne0ii ω ≠ ∅
73 r19.44zv ( ω ≠ ∅ → ( ∃ 𝑙 ∈ ω ( 𝑥 = ( ( 𝑚𝑔 𝑛 ) ⊼𝑔 ( 𝑜𝑔 𝑙 ) ) ∨ 𝑥 = ∀𝑔 𝑜 ( 𝑚𝑔 𝑛 ) ) ↔ ( ∃ 𝑙 ∈ ω 𝑥 = ( ( 𝑚𝑔 𝑛 ) ⊼𝑔 ( 𝑜𝑔 𝑙 ) ) ∨ 𝑥 = ∀𝑔 𝑜 ( 𝑚𝑔 𝑛 ) ) ) )
74 72 73 ax-mp ( ∃ 𝑙 ∈ ω ( 𝑥 = ( ( 𝑚𝑔 𝑛 ) ⊼𝑔 ( 𝑜𝑔 𝑙 ) ) ∨ 𝑥 = ∀𝑔 𝑜 ( 𝑚𝑔 𝑛 ) ) ↔ ( ∃ 𝑙 ∈ ω 𝑥 = ( ( 𝑚𝑔 𝑛 ) ⊼𝑔 ( 𝑜𝑔 𝑙 ) ) ∨ 𝑥 = ∀𝑔 𝑜 ( 𝑚𝑔 𝑛 ) ) )
75 eqeq1 ( 𝑦 = ( 𝑚𝑔 𝑛 ) → ( 𝑦 = ( 𝑖𝑔 𝑗 ) ↔ ( 𝑚𝑔 𝑛 ) = ( 𝑖𝑔 𝑗 ) ) )
76 75 2rexbidv ( 𝑦 = ( 𝑚𝑔 𝑛 ) → ( ∃ 𝑖 ∈ ω ∃ 𝑗 ∈ ω 𝑦 = ( 𝑖𝑔 𝑗 ) ↔ ∃ 𝑖 ∈ ω ∃ 𝑗 ∈ ω ( 𝑚𝑔 𝑛 ) = ( 𝑖𝑔 𝑗 ) ) )
77 ovexd ( ( ( ( ( 𝑚 ∈ ω ∧ 𝑛 ∈ ω ) ∧ 𝑜 ∈ ω ) ∧ 𝑙 ∈ ω ) ∧ ( 𝑥 = ( ( 𝑚𝑔 𝑛 ) ⊼𝑔 ( 𝑜𝑔 𝑙 ) ) ∨ 𝑥 = ∀𝑔 𝑜 ( 𝑚𝑔 𝑛 ) ) ) → ( 𝑚𝑔 𝑛 ) ∈ V )
78 simpl ( ( 𝑚 ∈ ω ∧ 𝑛 ∈ ω ) → 𝑚 ∈ ω )
79 43 eqeq2d ( 𝑖 = 𝑚 → ( ( 𝑚𝑔 𝑛 ) = ( 𝑖𝑔 𝑗 ) ↔ ( 𝑚𝑔 𝑛 ) = ( 𝑚𝑔 𝑗 ) ) )
80 79 rexbidv ( 𝑖 = 𝑚 → ( ∃ 𝑗 ∈ ω ( 𝑚𝑔 𝑛 ) = ( 𝑖𝑔 𝑗 ) ↔ ∃ 𝑗 ∈ ω ( 𝑚𝑔 𝑛 ) = ( 𝑚𝑔 𝑗 ) ) )
81 80 adantl ( ( ( 𝑚 ∈ ω ∧ 𝑛 ∈ ω ) ∧ 𝑖 = 𝑚 ) → ( ∃ 𝑗 ∈ ω ( 𝑚𝑔 𝑛 ) = ( 𝑖𝑔 𝑗 ) ↔ ∃ 𝑗 ∈ ω ( 𝑚𝑔 𝑛 ) = ( 𝑚𝑔 𝑗 ) ) )
82 simpr ( ( 𝑚 ∈ ω ∧ 𝑛 ∈ ω ) → 𝑛 ∈ ω )
83 52 eqeq2d ( 𝑗 = 𝑛 → ( ( 𝑚𝑔 𝑛 ) = ( 𝑚𝑔 𝑗 ) ↔ ( 𝑚𝑔 𝑛 ) = ( 𝑚𝑔 𝑛 ) ) )
84 83 adantl ( ( ( 𝑚 ∈ ω ∧ 𝑛 ∈ ω ) ∧ 𝑗 = 𝑛 ) → ( ( 𝑚𝑔 𝑛 ) = ( 𝑚𝑔 𝑗 ) ↔ ( 𝑚𝑔 𝑛 ) = ( 𝑚𝑔 𝑛 ) ) )
85 eqidd ( ( 𝑚 ∈ ω ∧ 𝑛 ∈ ω ) → ( 𝑚𝑔 𝑛 ) = ( 𝑚𝑔 𝑛 ) )
86 82 84 85 rspcedvd ( ( 𝑚 ∈ ω ∧ 𝑛 ∈ ω ) → ∃ 𝑗 ∈ ω ( 𝑚𝑔 𝑛 ) = ( 𝑚𝑔 𝑗 ) )
87 78 81 86 rspcedvd ( ( 𝑚 ∈ ω ∧ 𝑛 ∈ ω ) → ∃ 𝑖 ∈ ω ∃ 𝑗 ∈ ω ( 𝑚𝑔 𝑛 ) = ( 𝑖𝑔 𝑗 ) )
88 87 ad5ant12 ( ( ( ( ( 𝑚 ∈ ω ∧ 𝑛 ∈ ω ) ∧ 𝑜 ∈ ω ) ∧ 𝑙 ∈ ω ) ∧ ( 𝑥 = ( ( 𝑚𝑔 𝑛 ) ⊼𝑔 ( 𝑜𝑔 𝑙 ) ) ∨ 𝑥 = ∀𝑔 𝑜 ( 𝑚𝑔 𝑛 ) ) ) → ∃ 𝑖 ∈ ω ∃ 𝑗 ∈ ω ( 𝑚𝑔 𝑛 ) = ( 𝑖𝑔 𝑗 ) )
89 76 77 88 elrabd ( ( ( ( ( 𝑚 ∈ ω ∧ 𝑛 ∈ ω ) ∧ 𝑜 ∈ ω ) ∧ 𝑙 ∈ ω ) ∧ ( 𝑥 = ( ( 𝑚𝑔 𝑛 ) ⊼𝑔 ( 𝑜𝑔 𝑙 ) ) ∨ 𝑥 = ∀𝑔 𝑜 ( 𝑚𝑔 𝑛 ) ) ) → ( 𝑚𝑔 𝑛 ) ∈ { 𝑦 ∈ V ∣ ∃ 𝑖 ∈ ω ∃ 𝑗 ∈ ω 𝑦 = ( 𝑖𝑔 𝑗 ) } )
90 oveq1 ( 𝑝 = ( 𝑚𝑔 𝑛 ) → ( 𝑝𝑔 𝑞 ) = ( ( 𝑚𝑔 𝑛 ) ⊼𝑔 𝑞 ) )
91 90 eqeq2d ( 𝑝 = ( 𝑚𝑔 𝑛 ) → ( 𝑥 = ( 𝑝𝑔 𝑞 ) ↔ 𝑥 = ( ( 𝑚𝑔 𝑛 ) ⊼𝑔 𝑞 ) ) )
92 91 rexbidv ( 𝑝 = ( 𝑚𝑔 𝑛 ) → ( ∃ 𝑞 ∈ ( Fmla ‘ ∅ ) 𝑥 = ( 𝑝𝑔 𝑞 ) ↔ ∃ 𝑞 ∈ ( Fmla ‘ ∅ ) 𝑥 = ( ( 𝑚𝑔 𝑛 ) ⊼𝑔 𝑞 ) ) )
93 eqidd ( 𝑝 = ( 𝑚𝑔 𝑛 ) → 𝑘 = 𝑘 )
94 id ( 𝑝 = ( 𝑚𝑔 𝑛 ) → 𝑝 = ( 𝑚𝑔 𝑛 ) )
95 93 94 goaleq12d ( 𝑝 = ( 𝑚𝑔 𝑛 ) → ∀𝑔 𝑘 𝑝 = ∀𝑔 𝑘 ( 𝑚𝑔 𝑛 ) )
96 95 eqeq2d ( 𝑝 = ( 𝑚𝑔 𝑛 ) → ( 𝑥 = ∀𝑔 𝑘 𝑝𝑥 = ∀𝑔 𝑘 ( 𝑚𝑔 𝑛 ) ) )
97 96 rexbidv ( 𝑝 = ( 𝑚𝑔 𝑛 ) → ( ∃ 𝑘 ∈ ω 𝑥 = ∀𝑔 𝑘 𝑝 ↔ ∃ 𝑘 ∈ ω 𝑥 = ∀𝑔 𝑘 ( 𝑚𝑔 𝑛 ) ) )
98 92 97 orbi12d ( 𝑝 = ( 𝑚𝑔 𝑛 ) → ( ( ∃ 𝑞 ∈ ( Fmla ‘ ∅ ) 𝑥 = ( 𝑝𝑔 𝑞 ) ∨ ∃ 𝑘 ∈ ω 𝑥 = ∀𝑔 𝑘 𝑝 ) ↔ ( ∃ 𝑞 ∈ ( Fmla ‘ ∅ ) 𝑥 = ( ( 𝑚𝑔 𝑛 ) ⊼𝑔 𝑞 ) ∨ ∃ 𝑘 ∈ ω 𝑥 = ∀𝑔 𝑘 ( 𝑚𝑔 𝑛 ) ) ) )
99 98 adantl ( ( ( ( ( ( 𝑚 ∈ ω ∧ 𝑛 ∈ ω ) ∧ 𝑜 ∈ ω ) ∧ 𝑙 ∈ ω ) ∧ ( 𝑥 = ( ( 𝑚𝑔 𝑛 ) ⊼𝑔 ( 𝑜𝑔 𝑙 ) ) ∨ 𝑥 = ∀𝑔 𝑜 ( 𝑚𝑔 𝑛 ) ) ) ∧ 𝑝 = ( 𝑚𝑔 𝑛 ) ) → ( ( ∃ 𝑞 ∈ ( Fmla ‘ ∅ ) 𝑥 = ( 𝑝𝑔 𝑞 ) ∨ ∃ 𝑘 ∈ ω 𝑥 = ∀𝑔 𝑘 𝑝 ) ↔ ( ∃ 𝑞 ∈ ( Fmla ‘ ∅ ) 𝑥 = ( ( 𝑚𝑔 𝑛 ) ⊼𝑔 𝑞 ) ∨ ∃ 𝑘 ∈ ω 𝑥 = ∀𝑔 𝑘 ( 𝑚𝑔 𝑛 ) ) ) )
100 ovexd ( ( ( ( 𝑚 ∈ ω ∧ 𝑛 ∈ ω ) ∧ 𝑜 ∈ ω ) ∧ 𝑙 ∈ ω ) → ( 𝑜𝑔 𝑙 ) ∈ V )
101 simpr ( ( ( 𝑚 ∈ ω ∧ 𝑛 ∈ ω ) ∧ 𝑜 ∈ ω ) → 𝑜 ∈ ω )
102 101 adantr ( ( ( ( 𝑚 ∈ ω ∧ 𝑛 ∈ ω ) ∧ 𝑜 ∈ ω ) ∧ 𝑙 ∈ ω ) → 𝑜 ∈ ω )
103 oveq1 ( 𝑖 = 𝑜 → ( 𝑖𝑔 𝑗 ) = ( 𝑜𝑔 𝑗 ) )
104 103 eqeq2d ( 𝑖 = 𝑜 → ( ( 𝑜𝑔 𝑙 ) = ( 𝑖𝑔 𝑗 ) ↔ ( 𝑜𝑔 𝑙 ) = ( 𝑜𝑔 𝑗 ) ) )
105 104 rexbidv ( 𝑖 = 𝑜 → ( ∃ 𝑗 ∈ ω ( 𝑜𝑔 𝑙 ) = ( 𝑖𝑔 𝑗 ) ↔ ∃ 𝑗 ∈ ω ( 𝑜𝑔 𝑙 ) = ( 𝑜𝑔 𝑗 ) ) )
106 105 adantl ( ( ( ( ( 𝑚 ∈ ω ∧ 𝑛 ∈ ω ) ∧ 𝑜 ∈ ω ) ∧ 𝑙 ∈ ω ) ∧ 𝑖 = 𝑜 ) → ( ∃ 𝑗 ∈ ω ( 𝑜𝑔 𝑙 ) = ( 𝑖𝑔 𝑗 ) ↔ ∃ 𝑗 ∈ ω ( 𝑜𝑔 𝑙 ) = ( 𝑜𝑔 𝑗 ) ) )
107 simpr ( ( ( ( 𝑚 ∈ ω ∧ 𝑛 ∈ ω ) ∧ 𝑜 ∈ ω ) ∧ 𝑙 ∈ ω ) → 𝑙 ∈ ω )
108 oveq2 ( 𝑗 = 𝑙 → ( 𝑜𝑔 𝑗 ) = ( 𝑜𝑔 𝑙 ) )
109 108 eqeq2d ( 𝑗 = 𝑙 → ( ( 𝑜𝑔 𝑙 ) = ( 𝑜𝑔 𝑗 ) ↔ ( 𝑜𝑔 𝑙 ) = ( 𝑜𝑔 𝑙 ) ) )
110 109 adantl ( ( ( ( ( 𝑚 ∈ ω ∧ 𝑛 ∈ ω ) ∧ 𝑜 ∈ ω ) ∧ 𝑙 ∈ ω ) ∧ 𝑗 = 𝑙 ) → ( ( 𝑜𝑔 𝑙 ) = ( 𝑜𝑔 𝑗 ) ↔ ( 𝑜𝑔 𝑙 ) = ( 𝑜𝑔 𝑙 ) ) )
111 eqidd ( ( ( ( 𝑚 ∈ ω ∧ 𝑛 ∈ ω ) ∧ 𝑜 ∈ ω ) ∧ 𝑙 ∈ ω ) → ( 𝑜𝑔 𝑙 ) = ( 𝑜𝑔 𝑙 ) )
112 107 110 111 rspcedvd ( ( ( ( 𝑚 ∈ ω ∧ 𝑛 ∈ ω ) ∧ 𝑜 ∈ ω ) ∧ 𝑙 ∈ ω ) → ∃ 𝑗 ∈ ω ( 𝑜𝑔 𝑙 ) = ( 𝑜𝑔 𝑗 ) )
113 102 106 112 rspcedvd ( ( ( ( 𝑚 ∈ ω ∧ 𝑛 ∈ ω ) ∧ 𝑜 ∈ ω ) ∧ 𝑙 ∈ ω ) → ∃ 𝑖 ∈ ω ∃ 𝑗 ∈ ω ( 𝑜𝑔 𝑙 ) = ( 𝑖𝑔 𝑗 ) )
114 eqeq1 ( 𝑝 = ( 𝑜𝑔 𝑙 ) → ( 𝑝 = ( 𝑖𝑔 𝑗 ) ↔ ( 𝑜𝑔 𝑙 ) = ( 𝑖𝑔 𝑗 ) ) )
115 114 2rexbidv ( 𝑝 = ( 𝑜𝑔 𝑙 ) → ( ∃ 𝑖 ∈ ω ∃ 𝑗 ∈ ω 𝑝 = ( 𝑖𝑔 𝑗 ) ↔ ∃ 𝑖 ∈ ω ∃ 𝑗 ∈ ω ( 𝑜𝑔 𝑙 ) = ( 𝑖𝑔 𝑗 ) ) )
116 fmla0 ( Fmla ‘ ∅ ) = { 𝑝 ∈ V ∣ ∃ 𝑖 ∈ ω ∃ 𝑗 ∈ ω 𝑝 = ( 𝑖𝑔 𝑗 ) }
117 115 116 elrab2 ( ( 𝑜𝑔 𝑙 ) ∈ ( Fmla ‘ ∅ ) ↔ ( ( 𝑜𝑔 𝑙 ) ∈ V ∧ ∃ 𝑖 ∈ ω ∃ 𝑗 ∈ ω ( 𝑜𝑔 𝑙 ) = ( 𝑖𝑔 𝑗 ) ) )
118 100 113 117 sylanbrc ( ( ( ( 𝑚 ∈ ω ∧ 𝑛 ∈ ω ) ∧ 𝑜 ∈ ω ) ∧ 𝑙 ∈ ω ) → ( 𝑜𝑔 𝑙 ) ∈ ( Fmla ‘ ∅ ) )
119 118 adantr ( ( ( ( ( 𝑚 ∈ ω ∧ 𝑛 ∈ ω ) ∧ 𝑜 ∈ ω ) ∧ 𝑙 ∈ ω ) ∧ 𝑥 = ( ( 𝑚𝑔 𝑛 ) ⊼𝑔 ( 𝑜𝑔 𝑙 ) ) ) → ( 𝑜𝑔 𝑙 ) ∈ ( Fmla ‘ ∅ ) )
120 oveq2 ( 𝑞 = ( 𝑜𝑔 𝑙 ) → ( ( 𝑚𝑔 𝑛 ) ⊼𝑔 𝑞 ) = ( ( 𝑚𝑔 𝑛 ) ⊼𝑔 ( 𝑜𝑔 𝑙 ) ) )
121 120 eqeq2d ( 𝑞 = ( 𝑜𝑔 𝑙 ) → ( 𝑥 = ( ( 𝑚𝑔 𝑛 ) ⊼𝑔 𝑞 ) ↔ 𝑥 = ( ( 𝑚𝑔 𝑛 ) ⊼𝑔 ( 𝑜𝑔 𝑙 ) ) ) )
122 121 adantl ( ( ( ( ( ( 𝑚 ∈ ω ∧ 𝑛 ∈ ω ) ∧ 𝑜 ∈ ω ) ∧ 𝑙 ∈ ω ) ∧ 𝑥 = ( ( 𝑚𝑔 𝑛 ) ⊼𝑔 ( 𝑜𝑔 𝑙 ) ) ) ∧ 𝑞 = ( 𝑜𝑔 𝑙 ) ) → ( 𝑥 = ( ( 𝑚𝑔 𝑛 ) ⊼𝑔 𝑞 ) ↔ 𝑥 = ( ( 𝑚𝑔 𝑛 ) ⊼𝑔 ( 𝑜𝑔 𝑙 ) ) ) )
123 simpr ( ( ( ( ( 𝑚 ∈ ω ∧ 𝑛 ∈ ω ) ∧ 𝑜 ∈ ω ) ∧ 𝑙 ∈ ω ) ∧ 𝑥 = ( ( 𝑚𝑔 𝑛 ) ⊼𝑔 ( 𝑜𝑔 𝑙 ) ) ) → 𝑥 = ( ( 𝑚𝑔 𝑛 ) ⊼𝑔 ( 𝑜𝑔 𝑙 ) ) )
124 119 122 123 rspcedvd ( ( ( ( ( 𝑚 ∈ ω ∧ 𝑛 ∈ ω ) ∧ 𝑜 ∈ ω ) ∧ 𝑙 ∈ ω ) ∧ 𝑥 = ( ( 𝑚𝑔 𝑛 ) ⊼𝑔 ( 𝑜𝑔 𝑙 ) ) ) → ∃ 𝑞 ∈ ( Fmla ‘ ∅ ) 𝑥 = ( ( 𝑚𝑔 𝑛 ) ⊼𝑔 𝑞 ) )
125 124 ex ( ( ( ( 𝑚 ∈ ω ∧ 𝑛 ∈ ω ) ∧ 𝑜 ∈ ω ) ∧ 𝑙 ∈ ω ) → ( 𝑥 = ( ( 𝑚𝑔 𝑛 ) ⊼𝑔 ( 𝑜𝑔 𝑙 ) ) → ∃ 𝑞 ∈ ( Fmla ‘ ∅ ) 𝑥 = ( ( 𝑚𝑔 𝑛 ) ⊼𝑔 𝑞 ) ) )
126 102 adantr ( ( ( ( ( 𝑚 ∈ ω ∧ 𝑛 ∈ ω ) ∧ 𝑜 ∈ ω ) ∧ 𝑙 ∈ ω ) ∧ 𝑥 = ∀𝑔 𝑜 ( 𝑚𝑔 𝑛 ) ) → 𝑜 ∈ ω )
127 69 adantl ( ( ( ( ( ( 𝑚 ∈ ω ∧ 𝑛 ∈ ω ) ∧ 𝑜 ∈ ω ) ∧ 𝑙 ∈ ω ) ∧ 𝑥 = ∀𝑔 𝑜 ( 𝑚𝑔 𝑛 ) ) ∧ 𝑘 = 𝑜 ) → ( 𝑥 = ∀𝑔 𝑘 ( 𝑚𝑔 𝑛 ) ↔ 𝑥 = ∀𝑔 𝑜 ( 𝑚𝑔 𝑛 ) ) )
128 simpr ( ( ( ( ( 𝑚 ∈ ω ∧ 𝑛 ∈ ω ) ∧ 𝑜 ∈ ω ) ∧ 𝑙 ∈ ω ) ∧ 𝑥 = ∀𝑔 𝑜 ( 𝑚𝑔 𝑛 ) ) → 𝑥 = ∀𝑔 𝑜 ( 𝑚𝑔 𝑛 ) )
129 126 127 128 rspcedvd ( ( ( ( ( 𝑚 ∈ ω ∧ 𝑛 ∈ ω ) ∧ 𝑜 ∈ ω ) ∧ 𝑙 ∈ ω ) ∧ 𝑥 = ∀𝑔 𝑜 ( 𝑚𝑔 𝑛 ) ) → ∃ 𝑘 ∈ ω 𝑥 = ∀𝑔 𝑘 ( 𝑚𝑔 𝑛 ) )
130 129 ex ( ( ( ( 𝑚 ∈ ω ∧ 𝑛 ∈ ω ) ∧ 𝑜 ∈ ω ) ∧ 𝑙 ∈ ω ) → ( 𝑥 = ∀𝑔 𝑜 ( 𝑚𝑔 𝑛 ) → ∃ 𝑘 ∈ ω 𝑥 = ∀𝑔 𝑘 ( 𝑚𝑔 𝑛 ) ) )
131 125 130 orim12d ( ( ( ( 𝑚 ∈ ω ∧ 𝑛 ∈ ω ) ∧ 𝑜 ∈ ω ) ∧ 𝑙 ∈ ω ) → ( ( 𝑥 = ( ( 𝑚𝑔 𝑛 ) ⊼𝑔 ( 𝑜𝑔 𝑙 ) ) ∨ 𝑥 = ∀𝑔 𝑜 ( 𝑚𝑔 𝑛 ) ) → ( ∃ 𝑞 ∈ ( Fmla ‘ ∅ ) 𝑥 = ( ( 𝑚𝑔 𝑛 ) ⊼𝑔 𝑞 ) ∨ ∃ 𝑘 ∈ ω 𝑥 = ∀𝑔 𝑘 ( 𝑚𝑔 𝑛 ) ) ) )
132 131 imp ( ( ( ( ( 𝑚 ∈ ω ∧ 𝑛 ∈ ω ) ∧ 𝑜 ∈ ω ) ∧ 𝑙 ∈ ω ) ∧ ( 𝑥 = ( ( 𝑚𝑔 𝑛 ) ⊼𝑔 ( 𝑜𝑔 𝑙 ) ) ∨ 𝑥 = ∀𝑔 𝑜 ( 𝑚𝑔 𝑛 ) ) ) → ( ∃ 𝑞 ∈ ( Fmla ‘ ∅ ) 𝑥 = ( ( 𝑚𝑔 𝑛 ) ⊼𝑔 𝑞 ) ∨ ∃ 𝑘 ∈ ω 𝑥 = ∀𝑔 𝑘 ( 𝑚𝑔 𝑛 ) ) )
133 89 99 132 rspcedvd ( ( ( ( ( 𝑚 ∈ ω ∧ 𝑛 ∈ ω ) ∧ 𝑜 ∈ ω ) ∧ 𝑙 ∈ ω ) ∧ ( 𝑥 = ( ( 𝑚𝑔 𝑛 ) ⊼𝑔 ( 𝑜𝑔 𝑙 ) ) ∨ 𝑥 = ∀𝑔 𝑜 ( 𝑚𝑔 𝑛 ) ) ) → ∃ 𝑝 ∈ { 𝑦 ∈ V ∣ ∃ 𝑖 ∈ ω ∃ 𝑗 ∈ ω 𝑦 = ( 𝑖𝑔 𝑗 ) } ( ∃ 𝑞 ∈ ( Fmla ‘ ∅ ) 𝑥 = ( 𝑝𝑔 𝑞 ) ∨ ∃ 𝑘 ∈ ω 𝑥 = ∀𝑔 𝑘 𝑝 ) )
134 133 ex ( ( ( ( 𝑚 ∈ ω ∧ 𝑛 ∈ ω ) ∧ 𝑜 ∈ ω ) ∧ 𝑙 ∈ ω ) → ( ( 𝑥 = ( ( 𝑚𝑔 𝑛 ) ⊼𝑔 ( 𝑜𝑔 𝑙 ) ) ∨ 𝑥 = ∀𝑔 𝑜 ( 𝑚𝑔 𝑛 ) ) → ∃ 𝑝 ∈ { 𝑦 ∈ V ∣ ∃ 𝑖 ∈ ω ∃ 𝑗 ∈ ω 𝑦 = ( 𝑖𝑔 𝑗 ) } ( ∃ 𝑞 ∈ ( Fmla ‘ ∅ ) 𝑥 = ( 𝑝𝑔 𝑞 ) ∨ ∃ 𝑘 ∈ ω 𝑥 = ∀𝑔 𝑘 𝑝 ) ) )
135 134 rexlimdva ( ( ( 𝑚 ∈ ω ∧ 𝑛 ∈ ω ) ∧ 𝑜 ∈ ω ) → ( ∃ 𝑙 ∈ ω ( 𝑥 = ( ( 𝑚𝑔 𝑛 ) ⊼𝑔 ( 𝑜𝑔 𝑙 ) ) ∨ 𝑥 = ∀𝑔 𝑜 ( 𝑚𝑔 𝑛 ) ) → ∃ 𝑝 ∈ { 𝑦 ∈ V ∣ ∃ 𝑖 ∈ ω ∃ 𝑗 ∈ ω 𝑦 = ( 𝑖𝑔 𝑗 ) } ( ∃ 𝑞 ∈ ( Fmla ‘ ∅ ) 𝑥 = ( 𝑝𝑔 𝑞 ) ∨ ∃ 𝑘 ∈ ω 𝑥 = ∀𝑔 𝑘 𝑝 ) ) )
136 74 135 syl5bir ( ( ( 𝑚 ∈ ω ∧ 𝑛 ∈ ω ) ∧ 𝑜 ∈ ω ) → ( ( ∃ 𝑙 ∈ ω 𝑥 = ( ( 𝑚𝑔 𝑛 ) ⊼𝑔 ( 𝑜𝑔 𝑙 ) ) ∨ 𝑥 = ∀𝑔 𝑜 ( 𝑚𝑔 𝑛 ) ) → ∃ 𝑝 ∈ { 𝑦 ∈ V ∣ ∃ 𝑖 ∈ ω ∃ 𝑗 ∈ ω 𝑦 = ( 𝑖𝑔 𝑗 ) } ( ∃ 𝑞 ∈ ( Fmla ‘ ∅ ) 𝑥 = ( 𝑝𝑔 𝑞 ) ∨ ∃ 𝑘 ∈ ω 𝑥 = ∀𝑔 𝑘 𝑝 ) ) )
137 136 rexlimdva ( ( 𝑚 ∈ ω ∧ 𝑛 ∈ ω ) → ( ∃ 𝑜 ∈ ω ( ∃ 𝑙 ∈ ω 𝑥 = ( ( 𝑚𝑔 𝑛 ) ⊼𝑔 ( 𝑜𝑔 𝑙 ) ) ∨ 𝑥 = ∀𝑔 𝑜 ( 𝑚𝑔 𝑛 ) ) → ∃ 𝑝 ∈ { 𝑦 ∈ V ∣ ∃ 𝑖 ∈ ω ∃ 𝑗 ∈ ω 𝑦 = ( 𝑖𝑔 𝑗 ) } ( ∃ 𝑞 ∈ ( Fmla ‘ ∅ ) 𝑥 = ( 𝑝𝑔 𝑞 ) ∨ ∃ 𝑘 ∈ ω 𝑥 = ∀𝑔 𝑘 𝑝 ) ) )
138 71 137 syl5bi ( ( 𝑚 ∈ ω ∧ 𝑛 ∈ ω ) → ( ∃ 𝑘 ∈ ω ( ∃ 𝑙 ∈ ω 𝑥 = ( ( 𝑚𝑔 𝑛 ) ⊼𝑔 ( 𝑘𝑔 𝑙 ) ) ∨ 𝑥 = ∀𝑔 𝑘 ( 𝑚𝑔 𝑛 ) ) → ∃ 𝑝 ∈ { 𝑦 ∈ V ∣ ∃ 𝑖 ∈ ω ∃ 𝑗 ∈ ω 𝑦 = ( 𝑖𝑔 𝑗 ) } ( ∃ 𝑞 ∈ ( Fmla ‘ ∅ ) 𝑥 = ( 𝑝𝑔 𝑞 ) ∨ ∃ 𝑘 ∈ ω 𝑥 = ∀𝑔 𝑘 𝑝 ) ) )
139 138 rexlimivv ( ∃ 𝑚 ∈ ω ∃ 𝑛 ∈ ω ∃ 𝑘 ∈ ω ( ∃ 𝑙 ∈ ω 𝑥 = ( ( 𝑚𝑔 𝑛 ) ⊼𝑔 ( 𝑘𝑔 𝑙 ) ) ∨ 𝑥 = ∀𝑔 𝑘 ( 𝑚𝑔 𝑛 ) ) → ∃ 𝑝 ∈ { 𝑦 ∈ V ∣ ∃ 𝑖 ∈ ω ∃ 𝑗 ∈ ω 𝑦 = ( 𝑖𝑔 𝑗 ) } ( ∃ 𝑞 ∈ ( Fmla ‘ ∅ ) 𝑥 = ( 𝑝𝑔 𝑞 ) ∨ ∃ 𝑘 ∈ ω 𝑥 = ∀𝑔 𝑘 𝑝 ) )
140 61 139 sylbi ( ∃ 𝑖 ∈ ω ∃ 𝑗 ∈ ω ∃ 𝑘 ∈ ω ( ∃ 𝑙 ∈ ω 𝑥 = ( ( 𝑖𝑔 𝑗 ) ⊼𝑔 ( 𝑘𝑔 𝑙 ) ) ∨ 𝑥 = ∀𝑔 𝑘 ( 𝑖𝑔 𝑗 ) ) → ∃ 𝑝 ∈ { 𝑦 ∈ V ∣ ∃ 𝑖 ∈ ω ∃ 𝑗 ∈ ω 𝑦 = ( 𝑖𝑔 𝑗 ) } ( ∃ 𝑞 ∈ ( Fmla ‘ ∅ ) 𝑥 = ( 𝑝𝑔 𝑞 ) ∨ ∃ 𝑘 ∈ ω 𝑥 = ∀𝑔 𝑘 𝑝 ) )
141 42 140 impbii ( ∃ 𝑝 ∈ { 𝑦 ∈ V ∣ ∃ 𝑖 ∈ ω ∃ 𝑗 ∈ ω 𝑦 = ( 𝑖𝑔 𝑗 ) } ( ∃ 𝑞 ∈ ( Fmla ‘ ∅ ) 𝑥 = ( 𝑝𝑔 𝑞 ) ∨ ∃ 𝑘 ∈ ω 𝑥 = ∀𝑔 𝑘 𝑝 ) ↔ ∃ 𝑖 ∈ ω ∃ 𝑗 ∈ ω ∃ 𝑘 ∈ ω ( ∃ 𝑙 ∈ ω 𝑥 = ( ( 𝑖𝑔 𝑗 ) ⊼𝑔 ( 𝑘𝑔 𝑙 ) ) ∨ 𝑥 = ∀𝑔 𝑘 ( 𝑖𝑔 𝑗 ) ) )
142 8 141 bitri ( ∃ 𝑝 ∈ ( Fmla ‘ ∅ ) ( ∃ 𝑞 ∈ ( Fmla ‘ ∅ ) 𝑥 = ( 𝑝𝑔 𝑞 ) ∨ ∃ 𝑘 ∈ ω 𝑥 = ∀𝑔 𝑘 𝑝 ) ↔ ∃ 𝑖 ∈ ω ∃ 𝑗 ∈ ω ∃ 𝑘 ∈ ω ( ∃ 𝑙 ∈ ω 𝑥 = ( ( 𝑖𝑔 𝑗 ) ⊼𝑔 ( 𝑘𝑔 𝑙 ) ) ∨ 𝑥 = ∀𝑔 𝑘 ( 𝑖𝑔 𝑗 ) ) )
143 142 abbii { 𝑥 ∣ ∃ 𝑝 ∈ ( Fmla ‘ ∅ ) ( ∃ 𝑞 ∈ ( Fmla ‘ ∅ ) 𝑥 = ( 𝑝𝑔 𝑞 ) ∨ ∃ 𝑘 ∈ ω 𝑥 = ∀𝑔 𝑘 𝑝 ) } = { 𝑥 ∣ ∃ 𝑖 ∈ ω ∃ 𝑗 ∈ ω ∃ 𝑘 ∈ ω ( ∃ 𝑙 ∈ ω 𝑥 = ( ( 𝑖𝑔 𝑗 ) ⊼𝑔 ( 𝑘𝑔 𝑙 ) ) ∨ 𝑥 = ∀𝑔 𝑘 ( 𝑖𝑔 𝑗 ) ) }
144 6 143 uneq12i ( ( Fmla ‘ ∅ ) ∪ { 𝑥 ∣ ∃ 𝑝 ∈ ( Fmla ‘ ∅ ) ( ∃ 𝑞 ∈ ( Fmla ‘ ∅ ) 𝑥 = ( 𝑝𝑔 𝑞 ) ∨ ∃ 𝑘 ∈ ω 𝑥 = ∀𝑔 𝑘 𝑝 ) } ) = ( ( { ∅ } × ( ω × ω ) ) ∪ { 𝑥 ∣ ∃ 𝑖 ∈ ω ∃ 𝑗 ∈ ω ∃ 𝑘 ∈ ω ( ∃ 𝑙 ∈ ω 𝑥 = ( ( 𝑖𝑔 𝑗 ) ⊼𝑔 ( 𝑘𝑔 𝑙 ) ) ∨ 𝑥 = ∀𝑔 𝑘 ( 𝑖𝑔 𝑗 ) ) } )
145 2 5 144 3eqtri ( Fmla ‘ 1o ) = ( ( { ∅ } × ( ω × ω ) ) ∪ { 𝑥 ∣ ∃ 𝑖 ∈ ω ∃ 𝑗 ∈ ω ∃ 𝑘 ∈ ω ( ∃ 𝑙 ∈ ω 𝑥 = ( ( 𝑖𝑔 𝑗 ) ⊼𝑔 ( 𝑘𝑔 𝑙 ) ) ∨ 𝑥 = ∀𝑔 𝑘 ( 𝑖𝑔 𝑗 ) ) } )