Metamath Proof Explorer

Theorem br6

Description: Substitution for a six-place predicate. (Contributed by Scott Fenton, 4-Oct-2013) (Revised by Mario Carneiro, 3-May-2015)

Ref Expression
Hypotheses br6.1 ⊢ ( 𝑎 = 𝐴 → ( 𝜑 ↔ 𝜓 ) )
br6.2 ⊢ ( 𝑏 = 𝐵 → ( 𝜓 ↔ 𝜒 ) )
br6.3 ⊢ ( 𝑐 = 𝐶 → ( 𝜒 ↔ 𝜃 ) )
br6.4 ⊢ ( 𝑑 = 𝐷 → ( 𝜃 ↔ 𝜏 ) )
br6.5 ⊢ ( 𝑒 = 𝐸 → ( 𝜏 ↔ 𝜂 ) )
br6.6 ⊢ ( 𝑓 = 𝐹 → ( 𝜂 ↔ 𝜁 ) )
br6.7 ⊢ ( 𝑥 = 𝑋 → 𝑃 = 𝑄 )
br6.8 ⊢ 𝑅 = { ⟨ 𝑝 , 𝑞 ⟩ ∣ ∃ 𝑥 ∈ 𝑆 ∃ 𝑎 ∈ 𝑃 ∃ 𝑏 ∈ 𝑃 ∃ 𝑐 ∈ 𝑃 ∃ 𝑑 ∈ 𝑃 ∃ 𝑒 ∈ 𝑃 ∃ 𝑓 ∈ 𝑃 ( 𝑝 = ⟨ 𝑎 , ⟨ 𝑏 , 𝑐 ⟩ ⟩ ∧ 𝑞 = ⟨ 𝑑 , ⟨ 𝑒 , 𝑓 ⟩ ⟩ ∧ 𝜑 ) }
Assertion br6 ( ( 𝑋 ∈ 𝑆 ∧ ( 𝐴 ∈ 𝑄 ∧ 𝐵 ∈ 𝑄 ∧ 𝐶 ∈ 𝑄 ) ∧ ( 𝐷 ∈ 𝑄 ∧ 𝐸 ∈ 𝑄 ∧ 𝐹 ∈ 𝑄 ) ) → ( ⟨ 𝐴 , ⟨ 𝐵 , 𝐶 ⟩ ⟩ 𝑅 ⟨ 𝐷 , ⟨ 𝐸 , 𝐹 ⟩ ⟩ ↔ 𝜁 ) )

Proof

Step Hyp Ref Expression
1 br6.1 ⊢ ( 𝑎 = 𝐴 → ( 𝜑 ↔ 𝜓 ) )
2 br6.2 ⊢ ( 𝑏 = 𝐵 → ( 𝜓 ↔ 𝜒 ) )
3 br6.3 ⊢ ( 𝑐 = 𝐶 → ( 𝜒 ↔ 𝜃 ) )
4 br6.4 ⊢ ( 𝑑 = 𝐷 → ( 𝜃 ↔ 𝜏 ) )
5 br6.5 ⊢ ( 𝑒 = 𝐸 → ( 𝜏 ↔ 𝜂 ) )
6 br6.6 ⊢ ( 𝑓 = 𝐹 → ( 𝜂 ↔ 𝜁 ) )
7 br6.7 ⊢ ( 𝑥 = 𝑋 → 𝑃 = 𝑄 )
8 br6.8 ⊢ 𝑅 = { ⟨ 𝑝 , 𝑞 ⟩ ∣ ∃ 𝑥 ∈ 𝑆 ∃ 𝑎 ∈ 𝑃 ∃ 𝑏 ∈ 𝑃 ∃ 𝑐 ∈ 𝑃 ∃ 𝑑 ∈ 𝑃 ∃ 𝑒 ∈ 𝑃 ∃ 𝑓 ∈ 𝑃 ( 𝑝 = ⟨ 𝑎 , ⟨ 𝑏 , 𝑐 ⟩ ⟩ ∧ 𝑞 = ⟨ 𝑑 , ⟨ 𝑒 , 𝑓 ⟩ ⟩ ∧ 𝜑 ) }
9 opex ⊢ ⟨ 𝐴 , ⟨ 𝐵 , 𝐶 ⟩ ⟩ ∈ V
10 opex ⊢ ⟨ 𝐷 , ⟨ 𝐸 , 𝐹 ⟩ ⟩ ∈ V
11 eqeq1 ⊢ ( 𝑝 = ⟨ 𝐴 , ⟨ 𝐵 , 𝐶 ⟩ ⟩ → ( 𝑝 = ⟨ 𝑎 , ⟨ 𝑏 , 𝑐 ⟩ ⟩ ↔ ⟨ 𝐴 , ⟨ 𝐵 , 𝐶 ⟩ ⟩ = ⟨ 𝑎 , ⟨ 𝑏 , 𝑐 ⟩ ⟩ ) )
12 eqcom ⊢ ( ⟨ 𝐴 , ⟨ 𝐵 , 𝐶 ⟩ ⟩ = ⟨ 𝑎 , ⟨ 𝑏 , 𝑐 ⟩ ⟩ ↔ ⟨ 𝑎 , ⟨ 𝑏 , 𝑐 ⟩ ⟩ = ⟨ 𝐴 , ⟨ 𝐵 , 𝐶 ⟩ ⟩ )
13 11 12 bitrdi ⊢ ( 𝑝 = ⟨ 𝐴 , ⟨ 𝐵 , 𝐶 ⟩ ⟩ → ( 𝑝 = ⟨ 𝑎 , ⟨ 𝑏 , 𝑐 ⟩ ⟩ ↔ ⟨ 𝑎 , ⟨ 𝑏 , 𝑐 ⟩ ⟩ = ⟨ 𝐴 , ⟨ 𝐵 , 𝐶 ⟩ ⟩ ) )
14 13 3anbi1d ⊢ ( 𝑝 = ⟨ 𝐴 , ⟨ 𝐵 , 𝐶 ⟩ ⟩ → ( ( 𝑝 = ⟨ 𝑎 , ⟨ 𝑏 , 𝑐 ⟩ ⟩ ∧ 𝑞 = ⟨ 𝑑 , ⟨ 𝑒 , 𝑓 ⟩ ⟩ ∧ 𝜑 ) ↔ ( ⟨ 𝑎 , ⟨ 𝑏 , 𝑐 ⟩ ⟩ = ⟨ 𝐴 , ⟨ 𝐵 , 𝐶 ⟩ ⟩ ∧ 𝑞 = ⟨ 𝑑 , ⟨ 𝑒 , 𝑓 ⟩ ⟩ ∧ 𝜑 ) ) )
15 14 rexbidv ⊢ ( 𝑝 = ⟨ 𝐴 , ⟨ 𝐵 , 𝐶 ⟩ ⟩ → ( ∃ 𝑓 ∈ 𝑃 ( 𝑝 = ⟨ 𝑎 , ⟨ 𝑏 , 𝑐 ⟩ ⟩ ∧ 𝑞 = ⟨ 𝑑 , ⟨ 𝑒 , 𝑓 ⟩ ⟩ ∧ 𝜑 ) ↔ ∃ 𝑓 ∈ 𝑃 ( ⟨ 𝑎 , ⟨ 𝑏 , 𝑐 ⟩ ⟩ = ⟨ 𝐴 , ⟨ 𝐵 , 𝐶 ⟩ ⟩ ∧ 𝑞 = ⟨ 𝑑 , ⟨ 𝑒 , 𝑓 ⟩ ⟩ ∧ 𝜑 ) ) )
16 15 2rexbidv ⊢ ( 𝑝 = ⟨ 𝐴 , ⟨ 𝐵 , 𝐶 ⟩ ⟩ → ( ∃ 𝑑 ∈ 𝑃 ∃ 𝑒 ∈ 𝑃 ∃ 𝑓 ∈ 𝑃 ( 𝑝 = ⟨ 𝑎 , ⟨ 𝑏 , 𝑐 ⟩ ⟩ ∧ 𝑞 = ⟨ 𝑑 , ⟨ 𝑒 , 𝑓 ⟩ ⟩ ∧ 𝜑 ) ↔ ∃ 𝑑 ∈ 𝑃 ∃ 𝑒 ∈ 𝑃 ∃ 𝑓 ∈ 𝑃 ( ⟨ 𝑎 , ⟨ 𝑏 , 𝑐 ⟩ ⟩ = ⟨ 𝐴 , ⟨ 𝐵 , 𝐶 ⟩ ⟩ ∧ 𝑞 = ⟨ 𝑑 , ⟨ 𝑒 , 𝑓 ⟩ ⟩ ∧ 𝜑 ) ) )
17 16 2rexbidv ⊢ ( 𝑝 = ⟨ 𝐴 , ⟨ 𝐵 , 𝐶 ⟩ ⟩ → ( ∃ 𝑏 ∈ 𝑃 ∃ 𝑐 ∈ 𝑃 ∃ 𝑑 ∈ 𝑃 ∃ 𝑒 ∈ 𝑃 ∃ 𝑓 ∈ 𝑃 ( 𝑝 = ⟨ 𝑎 , ⟨ 𝑏 , 𝑐 ⟩ ⟩ ∧ 𝑞 = ⟨ 𝑑 , ⟨ 𝑒 , 𝑓 ⟩ ⟩ ∧ 𝜑 ) ↔ ∃ 𝑏 ∈ 𝑃 ∃ 𝑐 ∈ 𝑃 ∃ 𝑑 ∈ 𝑃 ∃ 𝑒 ∈ 𝑃 ∃ 𝑓 ∈ 𝑃 ( ⟨ 𝑎 , ⟨ 𝑏 , 𝑐 ⟩ ⟩ = ⟨ 𝐴 , ⟨ 𝐵 , 𝐶 ⟩ ⟩ ∧ 𝑞 = ⟨ 𝑑 , ⟨ 𝑒 , 𝑓 ⟩ ⟩ ∧ 𝜑 ) ) )
18 17 2rexbidv ⊢ ( 𝑝 = ⟨ 𝐴 , ⟨ 𝐵 , 𝐶 ⟩ ⟩ → ( ∃ 𝑥 ∈ 𝑆 ∃ 𝑎 ∈ 𝑃 ∃ 𝑏 ∈ 𝑃 ∃ 𝑐 ∈ 𝑃 ∃ 𝑑 ∈ 𝑃 ∃ 𝑒 ∈ 𝑃 ∃ 𝑓 ∈ 𝑃 ( 𝑝 = ⟨ 𝑎 , ⟨ 𝑏 , 𝑐 ⟩ ⟩ ∧ 𝑞 = ⟨ 𝑑 , ⟨ 𝑒 , 𝑓 ⟩ ⟩ ∧ 𝜑 ) ↔ ∃ 𝑥 ∈ 𝑆 ∃ 𝑎 ∈ 𝑃 ∃ 𝑏 ∈ 𝑃 ∃ 𝑐 ∈ 𝑃 ∃ 𝑑 ∈ 𝑃 ∃ 𝑒 ∈ 𝑃 ∃ 𝑓 ∈ 𝑃 ( ⟨ 𝑎 , ⟨ 𝑏 , 𝑐 ⟩ ⟩ = ⟨ 𝐴 , ⟨ 𝐵 , 𝐶 ⟩ ⟩ ∧ 𝑞 = ⟨ 𝑑 , ⟨ 𝑒 , 𝑓 ⟩ ⟩ ∧ 𝜑 ) ) )
19 eqeq1 ⊢ ( 𝑞 = ⟨ 𝐷 , ⟨ 𝐸 , 𝐹 ⟩ ⟩ → ( 𝑞 = ⟨ 𝑑 , ⟨ 𝑒 , 𝑓 ⟩ ⟩ ↔ ⟨ 𝐷 , ⟨ 𝐸 , 𝐹 ⟩ ⟩ = ⟨ 𝑑 , ⟨ 𝑒 , 𝑓 ⟩ ⟩ ) )
20 eqcom ⊢ ( ⟨ 𝐷 , ⟨ 𝐸 , 𝐹 ⟩ ⟩ = ⟨ 𝑑 , ⟨ 𝑒 , 𝑓 ⟩ ⟩ ↔ ⟨ 𝑑 , ⟨ 𝑒 , 𝑓 ⟩ ⟩ = ⟨ 𝐷 , ⟨ 𝐸 , 𝐹 ⟩ ⟩ )
21 19 20 bitrdi ⊢ ( 𝑞 = ⟨ 𝐷 , ⟨ 𝐸 , 𝐹 ⟩ ⟩ → ( 𝑞 = ⟨ 𝑑 , ⟨ 𝑒 , 𝑓 ⟩ ⟩ ↔ ⟨ 𝑑 , ⟨ 𝑒 , 𝑓 ⟩ ⟩ = ⟨ 𝐷 , ⟨ 𝐸 , 𝐹 ⟩ ⟩ ) )
22 21 3anbi2d ⊢ ( 𝑞 = ⟨ 𝐷 , ⟨ 𝐸 , 𝐹 ⟩ ⟩ → ( ( ⟨ 𝑎 , ⟨ 𝑏 , 𝑐 ⟩ ⟩ = ⟨ 𝐴 , ⟨ 𝐵 , 𝐶 ⟩ ⟩ ∧ 𝑞 = ⟨ 𝑑 , ⟨ 𝑒 , 𝑓 ⟩ ⟩ ∧ 𝜑 ) ↔ ( ⟨ 𝑎 , ⟨ 𝑏 , 𝑐 ⟩ ⟩ = ⟨ 𝐴 , ⟨ 𝐵 , 𝐶 ⟩ ⟩ ∧ ⟨ 𝑑 , ⟨ 𝑒 , 𝑓 ⟩ ⟩ = ⟨ 𝐷 , ⟨ 𝐸 , 𝐹 ⟩ ⟩ ∧ 𝜑 ) ) )
23 22 rexbidv ⊢ ( 𝑞 = ⟨ 𝐷 , ⟨ 𝐸 , 𝐹 ⟩ ⟩ → ( ∃ 𝑓 ∈ 𝑃 ( ⟨ 𝑎 , ⟨ 𝑏 , 𝑐 ⟩ ⟩ = ⟨ 𝐴 , ⟨ 𝐵 , 𝐶 ⟩ ⟩ ∧ 𝑞 = ⟨ 𝑑 , ⟨ 𝑒 , 𝑓 ⟩ ⟩ ∧ 𝜑 ) ↔ ∃ 𝑓 ∈ 𝑃 ( ⟨ 𝑎 , ⟨ 𝑏 , 𝑐 ⟩ ⟩ = ⟨ 𝐴 , ⟨ 𝐵 , 𝐶 ⟩ ⟩ ∧ ⟨ 𝑑 , ⟨ 𝑒 , 𝑓 ⟩ ⟩ = ⟨ 𝐷 , ⟨ 𝐸 , 𝐹 ⟩ ⟩ ∧ 𝜑 ) ) )
24 23 2rexbidv ⊢ ( 𝑞 = ⟨ 𝐷 , ⟨ 𝐸 , 𝐹 ⟩ ⟩ → ( ∃ 𝑑 ∈ 𝑃 ∃ 𝑒 ∈ 𝑃 ∃ 𝑓 ∈ 𝑃 ( ⟨ 𝑎 , ⟨ 𝑏 , 𝑐 ⟩ ⟩ = ⟨ 𝐴 , ⟨ 𝐵 , 𝐶 ⟩ ⟩ ∧ 𝑞 = ⟨ 𝑑 , ⟨ 𝑒 , 𝑓 ⟩ ⟩ ∧ 𝜑 ) ↔ ∃ 𝑑 ∈ 𝑃 ∃ 𝑒 ∈ 𝑃 ∃ 𝑓 ∈ 𝑃 ( ⟨ 𝑎 , ⟨ 𝑏 , 𝑐 ⟩ ⟩ = ⟨ 𝐴 , ⟨ 𝐵 , 𝐶 ⟩ ⟩ ∧ ⟨ 𝑑 , ⟨ 𝑒 , 𝑓 ⟩ ⟩ = ⟨ 𝐷 , ⟨ 𝐸 , 𝐹 ⟩ ⟩ ∧ 𝜑 ) ) )
25 24 2rexbidv ⊢ ( 𝑞 = ⟨ 𝐷 , ⟨ 𝐸 , 𝐹 ⟩ ⟩ → ( ∃ 𝑏 ∈ 𝑃 ∃ 𝑐 ∈ 𝑃 ∃ 𝑑 ∈ 𝑃 ∃ 𝑒 ∈ 𝑃 ∃ 𝑓 ∈ 𝑃 ( ⟨ 𝑎 , ⟨ 𝑏 , 𝑐 ⟩ ⟩ = ⟨ 𝐴 , ⟨ 𝐵 , 𝐶 ⟩ ⟩ ∧ 𝑞 = ⟨ 𝑑 , ⟨ 𝑒 , 𝑓 ⟩ ⟩ ∧ 𝜑 ) ↔ ∃ 𝑏 ∈ 𝑃 ∃ 𝑐 ∈ 𝑃 ∃ 𝑑 ∈ 𝑃 ∃ 𝑒 ∈ 𝑃 ∃ 𝑓 ∈ 𝑃 ( ⟨ 𝑎 , ⟨ 𝑏 , 𝑐 ⟩ ⟩ = ⟨ 𝐴 , ⟨ 𝐵 , 𝐶 ⟩ ⟩ ∧ ⟨ 𝑑 , ⟨ 𝑒 , 𝑓 ⟩ ⟩ = ⟨ 𝐷 , ⟨ 𝐸 , 𝐹 ⟩ ⟩ ∧ 𝜑 ) ) )
26 25 2rexbidv ⊢ ( 𝑞 = ⟨ 𝐷 , ⟨ 𝐸 , 𝐹 ⟩ ⟩ → ( ∃ 𝑥 ∈ 𝑆 ∃ 𝑎 ∈ 𝑃 ∃ 𝑏 ∈ 𝑃 ∃ 𝑐 ∈ 𝑃 ∃ 𝑑 ∈ 𝑃 ∃ 𝑒 ∈ 𝑃 ∃ 𝑓 ∈ 𝑃 ( ⟨ 𝑎 , ⟨ 𝑏 , 𝑐 ⟩ ⟩ = ⟨ 𝐴 , ⟨ 𝐵 , 𝐶 ⟩ ⟩ ∧ 𝑞 = ⟨ 𝑑 , ⟨ 𝑒 , 𝑓 ⟩ ⟩ ∧ 𝜑 ) ↔ ∃ 𝑥 ∈ 𝑆 ∃ 𝑎 ∈ 𝑃 ∃ 𝑏 ∈ 𝑃 ∃ 𝑐 ∈ 𝑃 ∃ 𝑑 ∈ 𝑃 ∃ 𝑒 ∈ 𝑃 ∃ 𝑓 ∈ 𝑃 ( ⟨ 𝑎 , ⟨ 𝑏 , 𝑐 ⟩ ⟩ = ⟨ 𝐴 , ⟨ 𝐵 , 𝐶 ⟩ ⟩ ∧ ⟨ 𝑑 , ⟨ 𝑒 , 𝑓 ⟩ ⟩ = ⟨ 𝐷 , ⟨ 𝐸 , 𝐹 ⟩ ⟩ ∧ 𝜑 ) ) )
27 9 10 18 26 8 brab ⊢ ( ⟨ 𝐴 , ⟨ 𝐵 , 𝐶 ⟩ ⟩ 𝑅 ⟨ 𝐷 , ⟨ 𝐸 , 𝐹 ⟩ ⟩ ↔ ∃ 𝑥 ∈ 𝑆 ∃ 𝑎 ∈ 𝑃 ∃ 𝑏 ∈ 𝑃 ∃ 𝑐 ∈ 𝑃 ∃ 𝑑 ∈ 𝑃 ∃ 𝑒 ∈ 𝑃 ∃ 𝑓 ∈ 𝑃 ( ⟨ 𝑎 , ⟨ 𝑏 , 𝑐 ⟩ ⟩ = ⟨ 𝐴 , ⟨ 𝐵 , 𝐶 ⟩ ⟩ ∧ ⟨ 𝑑 , ⟨ 𝑒 , 𝑓 ⟩ ⟩ = ⟨ 𝐷 , ⟨ 𝐸 , 𝐹 ⟩ ⟩ ∧ 𝜑 ) )
28 vex ⊢ 𝑎 ∈ V
29 opex ⊢ ⟨ 𝑏 , 𝑐 ⟩ ∈ V
30 28 29 opth ⊢ ( ⟨ 𝑎 , ⟨ 𝑏 , 𝑐 ⟩ ⟩ = ⟨ 𝐴 , ⟨ 𝐵 , 𝐶 ⟩ ⟩ ↔ ( 𝑎 = 𝐴 ∧ ⟨ 𝑏 , 𝑐 ⟩ = ⟨ 𝐵 , 𝐶 ⟩ ) )
31 vex ⊢ 𝑏 ∈ V
32 vex ⊢ 𝑐 ∈ V
33 31 32 opth ⊢ ( ⟨ 𝑏 , 𝑐 ⟩ = ⟨ 𝐵 , 𝐶 ⟩ ↔ ( 𝑏 = 𝐵 ∧ 𝑐 = 𝐶 ) )
34 2 3 sylan9bb ⊢ ( ( 𝑏 = 𝐵 ∧ 𝑐 = 𝐶 ) → ( 𝜓 ↔ 𝜃 ) )
35 33 34 sylbi ⊢ ( ⟨ 𝑏 , 𝑐 ⟩ = ⟨ 𝐵 , 𝐶 ⟩ → ( 𝜓 ↔ 𝜃 ) )
36 1 35 sylan9bb ⊢ ( ( 𝑎 = 𝐴 ∧ ⟨ 𝑏 , 𝑐 ⟩ = ⟨ 𝐵 , 𝐶 ⟩ ) → ( 𝜑 ↔ 𝜃 ) )
37 30 36 sylbi ⊢ ( ⟨ 𝑎 , ⟨ 𝑏 , 𝑐 ⟩ ⟩ = ⟨ 𝐴 , ⟨ 𝐵 , 𝐶 ⟩ ⟩ → ( 𝜑 ↔ 𝜃 ) )
38 vex ⊢ 𝑑 ∈ V
39 opex ⊢ ⟨ 𝑒 , 𝑓 ⟩ ∈ V
40 38 39 opth ⊢ ( ⟨ 𝑑 , ⟨ 𝑒 , 𝑓 ⟩ ⟩ = ⟨ 𝐷 , ⟨ 𝐸 , 𝐹 ⟩ ⟩ ↔ ( 𝑑 = 𝐷 ∧ ⟨ 𝑒 , 𝑓 ⟩ = ⟨ 𝐸 , 𝐹 ⟩ ) )
41 vex ⊢ 𝑒 ∈ V
42 vex ⊢ 𝑓 ∈ V
43 41 42 opth ⊢ ( ⟨ 𝑒 , 𝑓 ⟩ = ⟨ 𝐸 , 𝐹 ⟩ ↔ ( 𝑒 = 𝐸 ∧ 𝑓 = 𝐹 ) )
44 5 6 sylan9bb ⊢ ( ( 𝑒 = 𝐸 ∧ 𝑓 = 𝐹 ) → ( 𝜏 ↔ 𝜁 ) )
45 43 44 sylbi ⊢ ( ⟨ 𝑒 , 𝑓 ⟩ = ⟨ 𝐸 , 𝐹 ⟩ → ( 𝜏 ↔ 𝜁 ) )
46 4 45 sylan9bb ⊢ ( ( 𝑑 = 𝐷 ∧ ⟨ 𝑒 , 𝑓 ⟩ = ⟨ 𝐸 , 𝐹 ⟩ ) → ( 𝜃 ↔ 𝜁 ) )
47 40 46 sylbi ⊢ ( ⟨ 𝑑 , ⟨ 𝑒 , 𝑓 ⟩ ⟩ = ⟨ 𝐷 , ⟨ 𝐸 , 𝐹 ⟩ ⟩ → ( 𝜃 ↔ 𝜁 ) )
48 37 47 sylan9bb ⊢ ( ( ⟨ 𝑎 , ⟨ 𝑏 , 𝑐 ⟩ ⟩ = ⟨ 𝐴 , ⟨ 𝐵 , 𝐶 ⟩ ⟩ ∧ ⟨ 𝑑 , ⟨ 𝑒 , 𝑓 ⟩ ⟩ = ⟨ 𝐷 , ⟨ 𝐸 , 𝐹 ⟩ ⟩ ) → ( 𝜑 ↔ 𝜁 ) )
49 48 biimp3a ⊢ ( ( ⟨ 𝑎 , ⟨ 𝑏 , 𝑐 ⟩ ⟩ = ⟨ 𝐴 , ⟨ 𝐵 , 𝐶 ⟩ ⟩ ∧ ⟨ 𝑑 , ⟨ 𝑒 , 𝑓 ⟩ ⟩ = ⟨ 𝐷 , ⟨ 𝐸 , 𝐹 ⟩ ⟩ ∧ 𝜑 ) → 𝜁 )
50 49 a1i ⊢ ( ( ( ( ( ( 𝑋 ∈ 𝑆 ∧ ( 𝐴 ∈ 𝑄 ∧ 𝐵 ∈ 𝑄 ∧ 𝐶 ∈ 𝑄 ) ∧ ( 𝐷 ∈ 𝑄 ∧ 𝐸 ∈ 𝑄 ∧ 𝐹 ∈ 𝑄 ) ) ∧ ( 𝑥 ∈ 𝑆 ∧ 𝑎 ∈ 𝑃 ) ) ∧ ( 𝑏 ∈ 𝑃 ∧ 𝑐 ∈ 𝑃 ) ) ∧ ( 𝑑 ∈ 𝑃 ∧ 𝑒 ∈ 𝑃 ) ) ∧ 𝑓 ∈ 𝑃 ) → ( ( ⟨ 𝑎 , ⟨ 𝑏 , 𝑐 ⟩ ⟩ = ⟨ 𝐴 , ⟨ 𝐵 , 𝐶 ⟩ ⟩ ∧ ⟨ 𝑑 , ⟨ 𝑒 , 𝑓 ⟩ ⟩ = ⟨ 𝐷 , ⟨ 𝐸 , 𝐹 ⟩ ⟩ ∧ 𝜑 ) → 𝜁 ) )
51 50 rexlimdva ⊢ ( ( ( ( ( 𝑋 ∈ 𝑆 ∧ ( 𝐴 ∈ 𝑄 ∧ 𝐵 ∈ 𝑄 ∧ 𝐶 ∈ 𝑄 ) ∧ ( 𝐷 ∈ 𝑄 ∧ 𝐸 ∈ 𝑄 ∧ 𝐹 ∈ 𝑄 ) ) ∧ ( 𝑥 ∈ 𝑆 ∧ 𝑎 ∈ 𝑃 ) ) ∧ ( 𝑏 ∈ 𝑃 ∧ 𝑐 ∈ 𝑃 ) ) ∧ ( 𝑑 ∈ 𝑃 ∧ 𝑒 ∈ 𝑃 ) ) → ( ∃ 𝑓 ∈ 𝑃 ( ⟨ 𝑎 , ⟨ 𝑏 , 𝑐 ⟩ ⟩ = ⟨ 𝐴 , ⟨ 𝐵 , 𝐶 ⟩ ⟩ ∧ ⟨ 𝑑 , ⟨ 𝑒 , 𝑓 ⟩ ⟩ = ⟨ 𝐷 , ⟨ 𝐸 , 𝐹 ⟩ ⟩ ∧ 𝜑 ) → 𝜁 ) )
52 51 rexlimdvva ⊢ ( ( ( ( 𝑋 ∈ 𝑆 ∧ ( 𝐴 ∈ 𝑄 ∧ 𝐵 ∈ 𝑄 ∧ 𝐶 ∈ 𝑄 ) ∧ ( 𝐷 ∈ 𝑄 ∧ 𝐸 ∈ 𝑄 ∧ 𝐹 ∈ 𝑄 ) ) ∧ ( 𝑥 ∈ 𝑆 ∧ 𝑎 ∈ 𝑃 ) ) ∧ ( 𝑏 ∈ 𝑃 ∧ 𝑐 ∈ 𝑃 ) ) → ( ∃ 𝑑 ∈ 𝑃 ∃ 𝑒 ∈ 𝑃 ∃ 𝑓 ∈ 𝑃 ( ⟨ 𝑎 , ⟨ 𝑏 , 𝑐 ⟩ ⟩ = ⟨ 𝐴 , ⟨ 𝐵 , 𝐶 ⟩ ⟩ ∧ ⟨ 𝑑 , ⟨ 𝑒 , 𝑓 ⟩ ⟩ = ⟨ 𝐷 , ⟨ 𝐸 , 𝐹 ⟩ ⟩ ∧ 𝜑 ) → 𝜁 ) )
53 52 rexlimdvva ⊢ ( ( ( 𝑋 ∈ 𝑆 ∧ ( 𝐴 ∈ 𝑄 ∧ 𝐵 ∈ 𝑄 ∧ 𝐶 ∈ 𝑄 ) ∧ ( 𝐷 ∈ 𝑄 ∧ 𝐸 ∈ 𝑄 ∧ 𝐹 ∈ 𝑄 ) ) ∧ ( 𝑥 ∈ 𝑆 ∧ 𝑎 ∈ 𝑃 ) ) → ( ∃ 𝑏 ∈ 𝑃 ∃ 𝑐 ∈ 𝑃 ∃ 𝑑 ∈ 𝑃 ∃ 𝑒 ∈ 𝑃 ∃ 𝑓 ∈ 𝑃 ( ⟨ 𝑎 , ⟨ 𝑏 , 𝑐 ⟩ ⟩ = ⟨ 𝐴 , ⟨ 𝐵 , 𝐶 ⟩ ⟩ ∧ ⟨ 𝑑 , ⟨ 𝑒 , 𝑓 ⟩ ⟩ = ⟨ 𝐷 , ⟨ 𝐸 , 𝐹 ⟩ ⟩ ∧ 𝜑 ) → 𝜁 ) )
54 53 rexlimdvva ⊢ ( ( 𝑋 ∈ 𝑆 ∧ ( 𝐴 ∈ 𝑄 ∧ 𝐵 ∈ 𝑄 ∧ 𝐶 ∈ 𝑄 ) ∧ ( 𝐷 ∈ 𝑄 ∧ 𝐸 ∈ 𝑄 ∧ 𝐹 ∈ 𝑄 ) ) → ( ∃ 𝑥 ∈ 𝑆 ∃ 𝑎 ∈ 𝑃 ∃ 𝑏 ∈ 𝑃 ∃ 𝑐 ∈ 𝑃 ∃ 𝑑 ∈ 𝑃 ∃ 𝑒 ∈ 𝑃 ∃ 𝑓 ∈ 𝑃 ( ⟨ 𝑎 , ⟨ 𝑏 , 𝑐 ⟩ ⟩ = ⟨ 𝐴 , ⟨ 𝐵 , 𝐶 ⟩ ⟩ ∧ ⟨ 𝑑 , ⟨ 𝑒 , 𝑓 ⟩ ⟩ = ⟨ 𝐷 , ⟨ 𝐸 , 𝐹 ⟩ ⟩ ∧ 𝜑 ) → 𝜁 ) )
55 simpl1 ⊢ ( ( ( 𝑋 ∈ 𝑆 ∧ ( 𝐴 ∈ 𝑄 ∧ 𝐵 ∈ 𝑄 ∧ 𝐶 ∈ 𝑄 ) ∧ ( 𝐷 ∈ 𝑄 ∧ 𝐸 ∈ 𝑄 ∧ 𝐹 ∈ 𝑄 ) ) ∧ 𝜁 ) → 𝑋 ∈ 𝑆 )
56 simpl2 ⊢ ( ( ( 𝑋 ∈ 𝑆 ∧ ( 𝐴 ∈ 𝑄 ∧ 𝐵 ∈ 𝑄 ∧ 𝐶 ∈ 𝑄 ) ∧ ( 𝐷 ∈ 𝑄 ∧ 𝐸 ∈ 𝑄 ∧ 𝐹 ∈ 𝑄 ) ) ∧ 𝜁 ) → ( 𝐴 ∈ 𝑄 ∧ 𝐵 ∈ 𝑄 ∧ 𝐶 ∈ 𝑄 ) )
57 opeq1 ⊢ ( 𝑑 = 𝐷 → ⟨ 𝑑 , ⟨ 𝑒 , 𝑓 ⟩ ⟩ = ⟨ 𝐷 , ⟨ 𝑒 , 𝑓 ⟩ ⟩ )
58 57 eqeq1d ⊢ ( 𝑑 = 𝐷 → ( ⟨ 𝑑 , ⟨ 𝑒 , 𝑓 ⟩ ⟩ = ⟨ 𝐷 , ⟨ 𝐸 , 𝐹 ⟩ ⟩ ↔ ⟨ 𝐷 , ⟨ 𝑒 , 𝑓 ⟩ ⟩ = ⟨ 𝐷 , ⟨ 𝐸 , 𝐹 ⟩ ⟩ ) )
59 58 4 3anbi23d ⊢ ( 𝑑 = 𝐷 → ( ( ⟨ 𝐴 , ⟨ 𝐵 , 𝐶 ⟩ ⟩ = ⟨ 𝐴 , ⟨ 𝐵 , 𝐶 ⟩ ⟩ ∧ ⟨ 𝑑 , ⟨ 𝑒 , 𝑓 ⟩ ⟩ = ⟨ 𝐷 , ⟨ 𝐸 , 𝐹 ⟩ ⟩ ∧ 𝜃 ) ↔ ( ⟨ 𝐴 , ⟨ 𝐵 , 𝐶 ⟩ ⟩ = ⟨ 𝐴 , ⟨ 𝐵 , 𝐶 ⟩ ⟩ ∧ ⟨ 𝐷 , ⟨ 𝑒 , 𝑓 ⟩ ⟩ = ⟨ 𝐷 , ⟨ 𝐸 , 𝐹 ⟩ ⟩ ∧ 𝜏 ) ) )
60 opeq1 ⊢ ( 𝑒 = 𝐸 → ⟨ 𝑒 , 𝑓 ⟩ = ⟨ 𝐸 , 𝑓 ⟩ )
61 60 opeq2d ⊢ ( 𝑒 = 𝐸 → ⟨ 𝐷 , ⟨ 𝑒 , 𝑓 ⟩ ⟩ = ⟨ 𝐷 , ⟨ 𝐸 , 𝑓 ⟩ ⟩ )
62 61 eqeq1d ⊢ ( 𝑒 = 𝐸 → ( ⟨ 𝐷 , ⟨ 𝑒 , 𝑓 ⟩ ⟩ = ⟨ 𝐷 , ⟨ 𝐸 , 𝐹 ⟩ ⟩ ↔ ⟨ 𝐷 , ⟨ 𝐸 , 𝑓 ⟩ ⟩ = ⟨ 𝐷 , ⟨ 𝐸 , 𝐹 ⟩ ⟩ ) )
63 62 5 3anbi23d ⊢ ( 𝑒 = 𝐸 → ( ( ⟨ 𝐴 , ⟨ 𝐵 , 𝐶 ⟩ ⟩ = ⟨ 𝐴 , ⟨ 𝐵 , 𝐶 ⟩ ⟩ ∧ ⟨ 𝐷 , ⟨ 𝑒 , 𝑓 ⟩ ⟩ = ⟨ 𝐷 , ⟨ 𝐸 , 𝐹 ⟩ ⟩ ∧ 𝜏 ) ↔ ( ⟨ 𝐴 , ⟨ 𝐵 , 𝐶 ⟩ ⟩ = ⟨ 𝐴 , ⟨ 𝐵 , 𝐶 ⟩ ⟩ ∧ ⟨ 𝐷 , ⟨ 𝐸 , 𝑓 ⟩ ⟩ = ⟨ 𝐷 , ⟨ 𝐸 , 𝐹 ⟩ ⟩ ∧ 𝜂 ) ) )
64 opeq2 ⊢ ( 𝑓 = 𝐹 → ⟨ 𝐸 , 𝑓 ⟩ = ⟨ 𝐸 , 𝐹 ⟩ )
65 64 opeq2d ⊢ ( 𝑓 = 𝐹 → ⟨ 𝐷 , ⟨ 𝐸 , 𝑓 ⟩ ⟩ = ⟨ 𝐷 , ⟨ 𝐸 , 𝐹 ⟩ ⟩ )
66 65 eqeq1d ⊢ ( 𝑓 = 𝐹 → ( ⟨ 𝐷 , ⟨ 𝐸 , 𝑓 ⟩ ⟩ = ⟨ 𝐷 , ⟨ 𝐸 , 𝐹 ⟩ ⟩ ↔ ⟨ 𝐷 , ⟨ 𝐸 , 𝐹 ⟩ ⟩ = ⟨ 𝐷 , ⟨ 𝐸 , 𝐹 ⟩ ⟩ ) )
67 66 6 3anbi23d ⊢ ( 𝑓 = 𝐹 → ( ( ⟨ 𝐴 , ⟨ 𝐵 , 𝐶 ⟩ ⟩ = ⟨ 𝐴 , ⟨ 𝐵 , 𝐶 ⟩ ⟩ ∧ ⟨ 𝐷 , ⟨ 𝐸 , 𝑓 ⟩ ⟩ = ⟨ 𝐷 , ⟨ 𝐸 , 𝐹 ⟩ ⟩ ∧ 𝜂 ) ↔ ( ⟨ 𝐴 , ⟨ 𝐵 , 𝐶 ⟩ ⟩ = ⟨ 𝐴 , ⟨ 𝐵 , 𝐶 ⟩ ⟩ ∧ ⟨ 𝐷 , ⟨ 𝐸 , 𝐹 ⟩ ⟩ = ⟨ 𝐷 , ⟨ 𝐸 , 𝐹 ⟩ ⟩ ∧ 𝜁 ) ) )
68 eqid ⊢ ⟨ 𝐴 , ⟨ 𝐵 , 𝐶 ⟩ ⟩ = ⟨ 𝐴 , ⟨ 𝐵 , 𝐶 ⟩ ⟩
69 eqid ⊢ ⟨ 𝐷 , ⟨ 𝐸 , 𝐹 ⟩ ⟩ = ⟨ 𝐷 , ⟨ 𝐸 , 𝐹 ⟩ ⟩
70 68 69 pm3.2i ⊢ ( ⟨ 𝐴 , ⟨ 𝐵 , 𝐶 ⟩ ⟩ = ⟨ 𝐴 , ⟨ 𝐵 , 𝐶 ⟩ ⟩ ∧ ⟨ 𝐷 , ⟨ 𝐸 , 𝐹 ⟩ ⟩ = ⟨ 𝐷 , ⟨ 𝐸 , 𝐹 ⟩ ⟩ )
71 df-3an ⊢ ( ( ⟨ 𝐴 , ⟨ 𝐵 , 𝐶 ⟩ ⟩ = ⟨ 𝐴 , ⟨ 𝐵 , 𝐶 ⟩ ⟩ ∧ ⟨ 𝐷 , ⟨ 𝐸 , 𝐹 ⟩ ⟩ = ⟨ 𝐷 , ⟨ 𝐸 , 𝐹 ⟩ ⟩ ∧ 𝜁 ) ↔ ( ( ⟨ 𝐴 , ⟨ 𝐵 , 𝐶 ⟩ ⟩ = ⟨ 𝐴 , ⟨ 𝐵 , 𝐶 ⟩ ⟩ ∧ ⟨ 𝐷 , ⟨ 𝐸 , 𝐹 ⟩ ⟩ = ⟨ 𝐷 , ⟨ 𝐸 , 𝐹 ⟩ ⟩ ) ∧ 𝜁 ) )
72 70 71 mpbiran ⊢ ( ( ⟨ 𝐴 , ⟨ 𝐵 , 𝐶 ⟩ ⟩ = ⟨ 𝐴 , ⟨ 𝐵 , 𝐶 ⟩ ⟩ ∧ ⟨ 𝐷 , ⟨ 𝐸 , 𝐹 ⟩ ⟩ = ⟨ 𝐷 , ⟨ 𝐸 , 𝐹 ⟩ ⟩ ∧ 𝜁 ) ↔ 𝜁 )
73 67 72 bitrdi ⊢ ( 𝑓 = 𝐹 → ( ( ⟨ 𝐴 , ⟨ 𝐵 , 𝐶 ⟩ ⟩ = ⟨ 𝐴 , ⟨ 𝐵 , 𝐶 ⟩ ⟩ ∧ ⟨ 𝐷 , ⟨ 𝐸 , 𝑓 ⟩ ⟩ = ⟨ 𝐷 , ⟨ 𝐸 , 𝐹 ⟩ ⟩ ∧ 𝜂 ) ↔ 𝜁 ) )
74 59 63 73 rspc3ev ⊢ ( ( ( 𝐷 ∈ 𝑄 ∧ 𝐸 ∈ 𝑄 ∧ 𝐹 ∈ 𝑄 ) ∧ 𝜁 ) → ∃ 𝑑 ∈ 𝑄 ∃ 𝑒 ∈ 𝑄 ∃ 𝑓 ∈ 𝑄 ( ⟨ 𝐴 , ⟨ 𝐵 , 𝐶 ⟩ ⟩ = ⟨ 𝐴 , ⟨ 𝐵 , 𝐶 ⟩ ⟩ ∧ ⟨ 𝑑 , ⟨ 𝑒 , 𝑓 ⟩ ⟩ = ⟨ 𝐷 , ⟨ 𝐸 , 𝐹 ⟩ ⟩ ∧ 𝜃 ) )
75 74 3ad2antl3 ⊢ ( ( ( 𝑋 ∈ 𝑆 ∧ ( 𝐴 ∈ 𝑄 ∧ 𝐵 ∈ 𝑄 ∧ 𝐶 ∈ 𝑄 ) ∧ ( 𝐷 ∈ 𝑄 ∧ 𝐸 ∈ 𝑄 ∧ 𝐹 ∈ 𝑄 ) ) ∧ 𝜁 ) → ∃ 𝑑 ∈ 𝑄 ∃ 𝑒 ∈ 𝑄 ∃ 𝑓 ∈ 𝑄 ( ⟨ 𝐴 , ⟨ 𝐵 , 𝐶 ⟩ ⟩ = ⟨ 𝐴 , ⟨ 𝐵 , 𝐶 ⟩ ⟩ ∧ ⟨ 𝑑 , ⟨ 𝑒 , 𝑓 ⟩ ⟩ = ⟨ 𝐷 , ⟨ 𝐸 , 𝐹 ⟩ ⟩ ∧ 𝜃 ) )
76 opeq1 ⊢ ( 𝑎 = 𝐴 → ⟨ 𝑎 , ⟨ 𝑏 , 𝑐 ⟩ ⟩ = ⟨ 𝐴 , ⟨ 𝑏 , 𝑐 ⟩ ⟩ )
77 76 eqeq1d ⊢ ( 𝑎 = 𝐴 → ( ⟨ 𝑎 , ⟨ 𝑏 , 𝑐 ⟩ ⟩ = ⟨ 𝐴 , ⟨ 𝐵 , 𝐶 ⟩ ⟩ ↔ ⟨ 𝐴 , ⟨ 𝑏 , 𝑐 ⟩ ⟩ = ⟨ 𝐴 , ⟨ 𝐵 , 𝐶 ⟩ ⟩ ) )
78 77 1 3anbi13d ⊢ ( 𝑎 = 𝐴 → ( ( ⟨ 𝑎 , ⟨ 𝑏 , 𝑐 ⟩ ⟩ = ⟨ 𝐴 , ⟨ 𝐵 , 𝐶 ⟩ ⟩ ∧ ⟨ 𝑑 , ⟨ 𝑒 , 𝑓 ⟩ ⟩ = ⟨ 𝐷 , ⟨ 𝐸 , 𝐹 ⟩ ⟩ ∧ 𝜑 ) ↔ ( ⟨ 𝐴 , ⟨ 𝑏 , 𝑐 ⟩ ⟩ = ⟨ 𝐴 , ⟨ 𝐵 , 𝐶 ⟩ ⟩ ∧ ⟨ 𝑑 , ⟨ 𝑒 , 𝑓 ⟩ ⟩ = ⟨ 𝐷 , ⟨ 𝐸 , 𝐹 ⟩ ⟩ ∧ 𝜓 ) ) )
79 78 rexbidv ⊢ ( 𝑎 = 𝐴 → ( ∃ 𝑓 ∈ 𝑄 ( ⟨ 𝑎 , ⟨ 𝑏 , 𝑐 ⟩ ⟩ = ⟨ 𝐴 , ⟨ 𝐵 , 𝐶 ⟩ ⟩ ∧ ⟨ 𝑑 , ⟨ 𝑒 , 𝑓 ⟩ ⟩ = ⟨ 𝐷 , ⟨ 𝐸 , 𝐹 ⟩ ⟩ ∧ 𝜑 ) ↔ ∃ 𝑓 ∈ 𝑄 ( ⟨ 𝐴 , ⟨ 𝑏 , 𝑐 ⟩ ⟩ = ⟨ 𝐴 , ⟨ 𝐵 , 𝐶 ⟩ ⟩ ∧ ⟨ 𝑑 , ⟨ 𝑒 , 𝑓 ⟩ ⟩ = ⟨ 𝐷 , ⟨ 𝐸 , 𝐹 ⟩ ⟩ ∧ 𝜓 ) ) )
80 79 2rexbidv ⊢ ( 𝑎 = 𝐴 → ( ∃ 𝑑 ∈ 𝑄 ∃ 𝑒 ∈ 𝑄 ∃ 𝑓 ∈ 𝑄 ( ⟨ 𝑎 , ⟨ 𝑏 , 𝑐 ⟩ ⟩ = ⟨ 𝐴 , ⟨ 𝐵 , 𝐶 ⟩ ⟩ ∧ ⟨ 𝑑 , ⟨ 𝑒 , 𝑓 ⟩ ⟩ = ⟨ 𝐷 , ⟨ 𝐸 , 𝐹 ⟩ ⟩ ∧ 𝜑 ) ↔ ∃ 𝑑 ∈ 𝑄 ∃ 𝑒 ∈ 𝑄 ∃ 𝑓 ∈ 𝑄 ( ⟨ 𝐴 , ⟨ 𝑏 , 𝑐 ⟩ ⟩ = ⟨ 𝐴 , ⟨ 𝐵 , 𝐶 ⟩ ⟩ ∧ ⟨ 𝑑 , ⟨ 𝑒 , 𝑓 ⟩ ⟩ = ⟨ 𝐷 , ⟨ 𝐸 , 𝐹 ⟩ ⟩ ∧ 𝜓 ) ) )
81 opeq1 ⊢ ( 𝑏 = 𝐵 → ⟨ 𝑏 , 𝑐 ⟩ = ⟨ 𝐵 , 𝑐 ⟩ )
82 81 opeq2d ⊢ ( 𝑏 = 𝐵 → ⟨ 𝐴 , ⟨ 𝑏 , 𝑐 ⟩ ⟩ = ⟨ 𝐴 , ⟨ 𝐵 , 𝑐 ⟩ ⟩ )
83 82 eqeq1d ⊢ ( 𝑏 = 𝐵 → ( ⟨ 𝐴 , ⟨ 𝑏 , 𝑐 ⟩ ⟩ = ⟨ 𝐴 , ⟨ 𝐵 , 𝐶 ⟩ ⟩ ↔ ⟨ 𝐴 , ⟨ 𝐵 , 𝑐 ⟩ ⟩ = ⟨ 𝐴 , ⟨ 𝐵 , 𝐶 ⟩ ⟩ ) )
84 83 2 3anbi13d ⊢ ( 𝑏 = 𝐵 → ( ( ⟨ 𝐴 , ⟨ 𝑏 , 𝑐 ⟩ ⟩ = ⟨ 𝐴 , ⟨ 𝐵 , 𝐶 ⟩ ⟩ ∧ ⟨ 𝑑 , ⟨ 𝑒 , 𝑓 ⟩ ⟩ = ⟨ 𝐷 , ⟨ 𝐸 , 𝐹 ⟩ ⟩ ∧ 𝜓 ) ↔ ( ⟨ 𝐴 , ⟨ 𝐵 , 𝑐 ⟩ ⟩ = ⟨ 𝐴 , ⟨ 𝐵 , 𝐶 ⟩ ⟩ ∧ ⟨ 𝑑 , ⟨ 𝑒 , 𝑓 ⟩ ⟩ = ⟨ 𝐷 , ⟨ 𝐸 , 𝐹 ⟩ ⟩ ∧ 𝜒 ) ) )
85 84 rexbidv ⊢ ( 𝑏 = 𝐵 → ( ∃ 𝑓 ∈ 𝑄 ( ⟨ 𝐴 , ⟨ 𝑏 , 𝑐 ⟩ ⟩ = ⟨ 𝐴 , ⟨ 𝐵 , 𝐶 ⟩ ⟩ ∧ ⟨ 𝑑 , ⟨ 𝑒 , 𝑓 ⟩ ⟩ = ⟨ 𝐷 , ⟨ 𝐸 , 𝐹 ⟩ ⟩ ∧ 𝜓 ) ↔ ∃ 𝑓 ∈ 𝑄 ( ⟨ 𝐴 , ⟨ 𝐵 , 𝑐 ⟩ ⟩ = ⟨ 𝐴 , ⟨ 𝐵 , 𝐶 ⟩ ⟩ ∧ ⟨ 𝑑 , ⟨ 𝑒 , 𝑓 ⟩ ⟩ = ⟨ 𝐷 , ⟨ 𝐸 , 𝐹 ⟩ ⟩ ∧ 𝜒 ) ) )
86 85 2rexbidv ⊢ ( 𝑏 = 𝐵 → ( ∃ 𝑑 ∈ 𝑄 ∃ 𝑒 ∈ 𝑄 ∃ 𝑓 ∈ 𝑄 ( ⟨ 𝐴 , ⟨ 𝑏 , 𝑐 ⟩ ⟩ = ⟨ 𝐴 , ⟨ 𝐵 , 𝐶 ⟩ ⟩ ∧ ⟨ 𝑑 , ⟨ 𝑒 , 𝑓 ⟩ ⟩ = ⟨ 𝐷 , ⟨ 𝐸 , 𝐹 ⟩ ⟩ ∧ 𝜓 ) ↔ ∃ 𝑑 ∈ 𝑄 ∃ 𝑒 ∈ 𝑄 ∃ 𝑓 ∈ 𝑄 ( ⟨ 𝐴 , ⟨ 𝐵 , 𝑐 ⟩ ⟩ = ⟨ 𝐴 , ⟨ 𝐵 , 𝐶 ⟩ ⟩ ∧ ⟨ 𝑑 , ⟨ 𝑒 , 𝑓 ⟩ ⟩ = ⟨ 𝐷 , ⟨ 𝐸 , 𝐹 ⟩ ⟩ ∧ 𝜒 ) ) )
87 opeq2 ⊢ ( 𝑐 = 𝐶 → ⟨ 𝐵 , 𝑐 ⟩ = ⟨ 𝐵 , 𝐶 ⟩ )
88 87 opeq2d ⊢ ( 𝑐 = 𝐶 → ⟨ 𝐴 , ⟨ 𝐵 , 𝑐 ⟩ ⟩ = ⟨ 𝐴 , ⟨ 𝐵 , 𝐶 ⟩ ⟩ )
89 88 eqeq1d ⊢ ( 𝑐 = 𝐶 → ( ⟨ 𝐴 , ⟨ 𝐵 , 𝑐 ⟩ ⟩ = ⟨ 𝐴 , ⟨ 𝐵 , 𝐶 ⟩ ⟩ ↔ ⟨ 𝐴 , ⟨ 𝐵 , 𝐶 ⟩ ⟩ = ⟨ 𝐴 , ⟨ 𝐵 , 𝐶 ⟩ ⟩ ) )
90 89 3 3anbi13d ⊢ ( 𝑐 = 𝐶 → ( ( ⟨ 𝐴 , ⟨ 𝐵 , 𝑐 ⟩ ⟩ = ⟨ 𝐴 , ⟨ 𝐵 , 𝐶 ⟩ ⟩ ∧ ⟨ 𝑑 , ⟨ 𝑒 , 𝑓 ⟩ ⟩ = ⟨ 𝐷 , ⟨ 𝐸 , 𝐹 ⟩ ⟩ ∧ 𝜒 ) ↔ ( ⟨ 𝐴 , ⟨ 𝐵 , 𝐶 ⟩ ⟩ = ⟨ 𝐴 , ⟨ 𝐵 , 𝐶 ⟩ ⟩ ∧ ⟨ 𝑑 , ⟨ 𝑒 , 𝑓 ⟩ ⟩ = ⟨ 𝐷 , ⟨ 𝐸 , 𝐹 ⟩ ⟩ ∧ 𝜃 ) ) )
91 90 rexbidv ⊢ ( 𝑐 = 𝐶 → ( ∃ 𝑓 ∈ 𝑄 ( ⟨ 𝐴 , ⟨ 𝐵 , 𝑐 ⟩ ⟩ = ⟨ 𝐴 , ⟨ 𝐵 , 𝐶 ⟩ ⟩ ∧ ⟨ 𝑑 , ⟨ 𝑒 , 𝑓 ⟩ ⟩ = ⟨ 𝐷 , ⟨ 𝐸 , 𝐹 ⟩ ⟩ ∧ 𝜒 ) ↔ ∃ 𝑓 ∈ 𝑄 ( ⟨ 𝐴 , ⟨ 𝐵 , 𝐶 ⟩ ⟩ = ⟨ 𝐴 , ⟨ 𝐵 , 𝐶 ⟩ ⟩ ∧ ⟨ 𝑑 , ⟨ 𝑒 , 𝑓 ⟩ ⟩ = ⟨ 𝐷 , ⟨ 𝐸 , 𝐹 ⟩ ⟩ ∧ 𝜃 ) ) )
92 91 2rexbidv ⊢ ( 𝑐 = 𝐶 → ( ∃ 𝑑 ∈ 𝑄 ∃ 𝑒 ∈ 𝑄 ∃ 𝑓 ∈ 𝑄 ( ⟨ 𝐴 , ⟨ 𝐵 , 𝑐 ⟩ ⟩ = ⟨ 𝐴 , ⟨ 𝐵 , 𝐶 ⟩ ⟩ ∧ ⟨ 𝑑 , ⟨ 𝑒 , 𝑓 ⟩ ⟩ = ⟨ 𝐷 , ⟨ 𝐸 , 𝐹 ⟩ ⟩ ∧ 𝜒 ) ↔ ∃ 𝑑 ∈ 𝑄 ∃ 𝑒 ∈ 𝑄 ∃ 𝑓 ∈ 𝑄 ( ⟨ 𝐴 , ⟨ 𝐵 , 𝐶 ⟩ ⟩ = ⟨ 𝐴 , ⟨ 𝐵 , 𝐶 ⟩ ⟩ ∧ ⟨ 𝑑 , ⟨ 𝑒 , 𝑓 ⟩ ⟩ = ⟨ 𝐷 , ⟨ 𝐸 , 𝐹 ⟩ ⟩ ∧ 𝜃 ) ) )
93 80 86 92 rspc3ev ⊢ ( ( ( 𝐴 ∈ 𝑄 ∧ 𝐵 ∈ 𝑄 ∧ 𝐶 ∈ 𝑄 ) ∧ ∃ 𝑑 ∈ 𝑄 ∃ 𝑒 ∈ 𝑄 ∃ 𝑓 ∈ 𝑄 ( ⟨ 𝐴 , ⟨ 𝐵 , 𝐶 ⟩ ⟩ = ⟨ 𝐴 , ⟨ 𝐵 , 𝐶 ⟩ ⟩ ∧ ⟨ 𝑑 , ⟨ 𝑒 , 𝑓 ⟩ ⟩ = ⟨ 𝐷 , ⟨ 𝐸 , 𝐹 ⟩ ⟩ ∧ 𝜃 ) ) → ∃ 𝑎 ∈ 𝑄 ∃ 𝑏 ∈ 𝑄 ∃ 𝑐 ∈ 𝑄 ∃ 𝑑 ∈ 𝑄 ∃ 𝑒 ∈ 𝑄 ∃ 𝑓 ∈ 𝑄 ( ⟨ 𝑎 , ⟨ 𝑏 , 𝑐 ⟩ ⟩ = ⟨ 𝐴 , ⟨ 𝐵 , 𝐶 ⟩ ⟩ ∧ ⟨ 𝑑 , ⟨ 𝑒 , 𝑓 ⟩ ⟩ = ⟨ 𝐷 , ⟨ 𝐸 , 𝐹 ⟩ ⟩ ∧ 𝜑 ) )
94 56 75 93 syl2anc ⊢ ( ( ( 𝑋 ∈ 𝑆 ∧ ( 𝐴 ∈ 𝑄 ∧ 𝐵 ∈ 𝑄 ∧ 𝐶 ∈ 𝑄 ) ∧ ( 𝐷 ∈ 𝑄 ∧ 𝐸 ∈ 𝑄 ∧ 𝐹 ∈ 𝑄 ) ) ∧ 𝜁 ) → ∃ 𝑎 ∈ 𝑄 ∃ 𝑏 ∈ 𝑄 ∃ 𝑐 ∈ 𝑄 ∃ 𝑑 ∈ 𝑄 ∃ 𝑒 ∈ 𝑄 ∃ 𝑓 ∈ 𝑄 ( ⟨ 𝑎 , ⟨ 𝑏 , 𝑐 ⟩ ⟩ = ⟨ 𝐴 , ⟨ 𝐵 , 𝐶 ⟩ ⟩ ∧ ⟨ 𝑑 , ⟨ 𝑒 , 𝑓 ⟩ ⟩ = ⟨ 𝐷 , ⟨ 𝐸 , 𝐹 ⟩ ⟩ ∧ 𝜑 ) )
95 7 rexeqdv ⊢ ( 𝑥 = 𝑋 → ( ∃ 𝑓 ∈ 𝑃 ( ⟨ 𝑎 , ⟨ 𝑏 , 𝑐 ⟩ ⟩ = ⟨ 𝐴 , ⟨ 𝐵 , 𝐶 ⟩ ⟩ ∧ ⟨ 𝑑 , ⟨ 𝑒 , 𝑓 ⟩ ⟩ = ⟨ 𝐷 , ⟨ 𝐸 , 𝐹 ⟩ ⟩ ∧ 𝜑 ) ↔ ∃ 𝑓 ∈ 𝑄 ( ⟨ 𝑎 , ⟨ 𝑏 , 𝑐 ⟩ ⟩ = ⟨ 𝐴 , ⟨ 𝐵 , 𝐶 ⟩ ⟩ ∧ ⟨ 𝑑 , ⟨ 𝑒 , 𝑓 ⟩ ⟩ = ⟨ 𝐷 , ⟨ 𝐸 , 𝐹 ⟩ ⟩ ∧ 𝜑 ) ) )
96 7 95 rexeqbidv ⊢ ( 𝑥 = 𝑋 → ( ∃ 𝑒 ∈ 𝑃 ∃ 𝑓 ∈ 𝑃 ( ⟨ 𝑎 , ⟨ 𝑏 , 𝑐 ⟩ ⟩ = ⟨ 𝐴 , ⟨ 𝐵 , 𝐶 ⟩ ⟩ ∧ ⟨ 𝑑 , ⟨ 𝑒 , 𝑓 ⟩ ⟩ = ⟨ 𝐷 , ⟨ 𝐸 , 𝐹 ⟩ ⟩ ∧ 𝜑 ) ↔ ∃ 𝑒 ∈ 𝑄 ∃ 𝑓 ∈ 𝑄 ( ⟨ 𝑎 , ⟨ 𝑏 , 𝑐 ⟩ ⟩ = ⟨ 𝐴 , ⟨ 𝐵 , 𝐶 ⟩ ⟩ ∧ ⟨ 𝑑 , ⟨ 𝑒 , 𝑓 ⟩ ⟩ = ⟨ 𝐷 , ⟨ 𝐸 , 𝐹 ⟩ ⟩ ∧ 𝜑 ) ) )
97 7 96 rexeqbidv ⊢ ( 𝑥 = 𝑋 → ( ∃ 𝑑 ∈ 𝑃 ∃ 𝑒 ∈ 𝑃 ∃ 𝑓 ∈ 𝑃 ( ⟨ 𝑎 , ⟨ 𝑏 , 𝑐 ⟩ ⟩ = ⟨ 𝐴 , ⟨ 𝐵 , 𝐶 ⟩ ⟩ ∧ ⟨ 𝑑 , ⟨ 𝑒 , 𝑓 ⟩ ⟩ = ⟨ 𝐷 , ⟨ 𝐸 , 𝐹 ⟩ ⟩ ∧ 𝜑 ) ↔ ∃ 𝑑 ∈ 𝑄 ∃ 𝑒 ∈ 𝑄 ∃ 𝑓 ∈ 𝑄 ( ⟨ 𝑎 , ⟨ 𝑏 , 𝑐 ⟩ ⟩ = ⟨ 𝐴 , ⟨ 𝐵 , 𝐶 ⟩ ⟩ ∧ ⟨ 𝑑 , ⟨ 𝑒 , 𝑓 ⟩ ⟩ = ⟨ 𝐷 , ⟨ 𝐸 , 𝐹 ⟩ ⟩ ∧ 𝜑 ) ) )
98 7 97 rexeqbidv ⊢ ( 𝑥 = 𝑋 → ( ∃ 𝑐 ∈ 𝑃 ∃ 𝑑 ∈ 𝑃 ∃ 𝑒 ∈ 𝑃 ∃ 𝑓 ∈ 𝑃 ( ⟨ 𝑎 , ⟨ 𝑏 , 𝑐 ⟩ ⟩ = ⟨ 𝐴 , ⟨ 𝐵 , 𝐶 ⟩ ⟩ ∧ ⟨ 𝑑 , ⟨ 𝑒 , 𝑓 ⟩ ⟩ = ⟨ 𝐷 , ⟨ 𝐸 , 𝐹 ⟩ ⟩ ∧ 𝜑 ) ↔ ∃ 𝑐 ∈ 𝑄 ∃ 𝑑 ∈ 𝑄 ∃ 𝑒 ∈ 𝑄 ∃ 𝑓 ∈ 𝑄 ( ⟨ 𝑎 , ⟨ 𝑏 , 𝑐 ⟩ ⟩ = ⟨ 𝐴 , ⟨ 𝐵 , 𝐶 ⟩ ⟩ ∧ ⟨ 𝑑 , ⟨ 𝑒 , 𝑓 ⟩ ⟩ = ⟨ 𝐷 , ⟨ 𝐸 , 𝐹 ⟩ ⟩ ∧ 𝜑 ) ) )
99 7 98 rexeqbidv ⊢ ( 𝑥 = 𝑋 → ( ∃ 𝑏 ∈ 𝑃 ∃ 𝑐 ∈ 𝑃 ∃ 𝑑 ∈ 𝑃 ∃ 𝑒 ∈ 𝑃 ∃ 𝑓 ∈ 𝑃 ( ⟨ 𝑎 , ⟨ 𝑏 , 𝑐 ⟩ ⟩ = ⟨ 𝐴 , ⟨ 𝐵 , 𝐶 ⟩ ⟩ ∧ ⟨ 𝑑 , ⟨ 𝑒 , 𝑓 ⟩ ⟩ = ⟨ 𝐷 , ⟨ 𝐸 , 𝐹 ⟩ ⟩ ∧ 𝜑 ) ↔ ∃ 𝑏 ∈ 𝑄 ∃ 𝑐 ∈ 𝑄 ∃ 𝑑 ∈ 𝑄 ∃ 𝑒 ∈ 𝑄 ∃ 𝑓 ∈ 𝑄 ( ⟨ 𝑎 , ⟨ 𝑏 , 𝑐 ⟩ ⟩ = ⟨ 𝐴 , ⟨ 𝐵 , 𝐶 ⟩ ⟩ ∧ ⟨ 𝑑 , ⟨ 𝑒 , 𝑓 ⟩ ⟩ = ⟨ 𝐷 , ⟨ 𝐸 , 𝐹 ⟩ ⟩ ∧ 𝜑 ) ) )
100 7 99 rexeqbidv ⊢ ( 𝑥 = 𝑋 → ( ∃ 𝑎 ∈ 𝑃 ∃ 𝑏 ∈ 𝑃 ∃ 𝑐 ∈ 𝑃 ∃ 𝑑 ∈ 𝑃 ∃ 𝑒 ∈ 𝑃 ∃ 𝑓 ∈ 𝑃 ( ⟨ 𝑎 , ⟨ 𝑏 , 𝑐 ⟩ ⟩ = ⟨ 𝐴 , ⟨ 𝐵 , 𝐶 ⟩ ⟩ ∧ ⟨ 𝑑 , ⟨ 𝑒 , 𝑓 ⟩ ⟩ = ⟨ 𝐷 , ⟨ 𝐸 , 𝐹 ⟩ ⟩ ∧ 𝜑 ) ↔ ∃ 𝑎 ∈ 𝑄 ∃ 𝑏 ∈ 𝑄 ∃ 𝑐 ∈ 𝑄 ∃ 𝑑 ∈ 𝑄 ∃ 𝑒 ∈ 𝑄 ∃ 𝑓 ∈ 𝑄 ( ⟨ 𝑎 , ⟨ 𝑏 , 𝑐 ⟩ ⟩ = ⟨ 𝐴 , ⟨ 𝐵 , 𝐶 ⟩ ⟩ ∧ ⟨ 𝑑 , ⟨ 𝑒 , 𝑓 ⟩ ⟩ = ⟨ 𝐷 , ⟨ 𝐸 , 𝐹 ⟩ ⟩ ∧ 𝜑 ) ) )
101 100 rspcev ⊢ ( ( 𝑋 ∈ 𝑆 ∧ ∃ 𝑎 ∈ 𝑄 ∃ 𝑏 ∈ 𝑄 ∃ 𝑐 ∈ 𝑄 ∃ 𝑑 ∈ 𝑄 ∃ 𝑒 ∈ 𝑄 ∃ 𝑓 ∈ 𝑄 ( ⟨ 𝑎 , ⟨ 𝑏 , 𝑐 ⟩ ⟩ = ⟨ 𝐴 , ⟨ 𝐵 , 𝐶 ⟩ ⟩ ∧ ⟨ 𝑑 , ⟨ 𝑒 , 𝑓 ⟩ ⟩ = ⟨ 𝐷 , ⟨ 𝐸 , 𝐹 ⟩ ⟩ ∧ 𝜑 ) ) → ∃ 𝑥 ∈ 𝑆 ∃ 𝑎 ∈ 𝑃 ∃ 𝑏 ∈ 𝑃 ∃ 𝑐 ∈ 𝑃 ∃ 𝑑 ∈ 𝑃 ∃ 𝑒 ∈ 𝑃 ∃ 𝑓 ∈ 𝑃 ( ⟨ 𝑎 , ⟨ 𝑏 , 𝑐 ⟩ ⟩ = ⟨ 𝐴 , ⟨ 𝐵 , 𝐶 ⟩ ⟩ ∧ ⟨ 𝑑 , ⟨ 𝑒 , 𝑓 ⟩ ⟩ = ⟨ 𝐷 , ⟨ 𝐸 , 𝐹 ⟩ ⟩ ∧ 𝜑 ) )
102 55 94 101 syl2anc ⊢ ( ( ( 𝑋 ∈ 𝑆 ∧ ( 𝐴 ∈ 𝑄 ∧ 𝐵 ∈ 𝑄 ∧ 𝐶 ∈ 𝑄 ) ∧ ( 𝐷 ∈ 𝑄 ∧ 𝐸 ∈ 𝑄 ∧ 𝐹 ∈ 𝑄 ) ) ∧ 𝜁 ) → ∃ 𝑥 ∈ 𝑆 ∃ 𝑎 ∈ 𝑃 ∃ 𝑏 ∈ 𝑃 ∃ 𝑐 ∈ 𝑃 ∃ 𝑑 ∈ 𝑃 ∃ 𝑒 ∈ 𝑃 ∃ 𝑓 ∈ 𝑃 ( ⟨ 𝑎 , ⟨ 𝑏 , 𝑐 ⟩ ⟩ = ⟨ 𝐴 , ⟨ 𝐵 , 𝐶 ⟩ ⟩ ∧ ⟨ 𝑑 , ⟨ 𝑒 , 𝑓 ⟩ ⟩ = ⟨ 𝐷 , ⟨ 𝐸 , 𝐹 ⟩ ⟩ ∧ 𝜑 ) )
103 102 ex ⊢ ( ( 𝑋 ∈ 𝑆 ∧ ( 𝐴 ∈ 𝑄 ∧ 𝐵 ∈ 𝑄 ∧ 𝐶 ∈ 𝑄 ) ∧ ( 𝐷 ∈ 𝑄 ∧ 𝐸 ∈ 𝑄 ∧ 𝐹 ∈ 𝑄 ) ) → ( 𝜁 → ∃ 𝑥 ∈ 𝑆 ∃ 𝑎 ∈ 𝑃 ∃ 𝑏 ∈ 𝑃 ∃ 𝑐 ∈ 𝑃 ∃ 𝑑 ∈ 𝑃 ∃ 𝑒 ∈ 𝑃 ∃ 𝑓 ∈ 𝑃 ( ⟨ 𝑎 , ⟨ 𝑏 , 𝑐 ⟩ ⟩ = ⟨ 𝐴 , ⟨ 𝐵 , 𝐶 ⟩ ⟩ ∧ ⟨ 𝑑 , ⟨ 𝑒 , 𝑓 ⟩ ⟩ = ⟨ 𝐷 , ⟨ 𝐸 , 𝐹 ⟩ ⟩ ∧ 𝜑 ) ) )
104 54 103 impbid ⊢ ( ( 𝑋 ∈ 𝑆 ∧ ( 𝐴 ∈ 𝑄 ∧ 𝐵 ∈ 𝑄 ∧ 𝐶 ∈ 𝑄 ) ∧ ( 𝐷 ∈ 𝑄 ∧ 𝐸 ∈ 𝑄 ∧ 𝐹 ∈ 𝑄 ) ) → ( ∃ 𝑥 ∈ 𝑆 ∃ 𝑎 ∈ 𝑃 ∃ 𝑏 ∈ 𝑃 ∃ 𝑐 ∈ 𝑃 ∃ 𝑑 ∈ 𝑃 ∃ 𝑒 ∈ 𝑃 ∃ 𝑓 ∈ 𝑃 ( ⟨ 𝑎 , ⟨ 𝑏 , 𝑐 ⟩ ⟩ = ⟨ 𝐴 , ⟨ 𝐵 , 𝐶 ⟩ ⟩ ∧ ⟨ 𝑑 , ⟨ 𝑒 , 𝑓 ⟩ ⟩ = ⟨ 𝐷 , ⟨ 𝐸 , 𝐹 ⟩ ⟩ ∧ 𝜑 ) ↔ 𝜁 ) )
105 27 104 syl5bb ⊢ ( ( 𝑋 ∈ 𝑆 ∧ ( 𝐴 ∈ 𝑄 ∧ 𝐵 ∈ 𝑄 ∧ 𝐶 ∈ 𝑄 ) ∧ ( 𝐷 ∈ 𝑄 ∧ 𝐸 ∈ 𝑄 ∧ 𝐹 ∈ 𝑄 ) ) → ( ⟨ 𝐴 , ⟨ 𝐵 , 𝐶 ⟩ ⟩ 𝑅 ⟨ 𝐷 , ⟨ 𝐸 , 𝐹 ⟩ ⟩ ↔ 𝜁 ) )