br8 - Mathbox for Scott Fenton

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Theorem br8 36500

Description: Substitution for an eight-place predicate. (Contributed by Scott Fenton, 26-Sep-2013.) (Revised by Mario Carneiro, 3-May-2015.)

**Hypotheses**
Ref	Expression
br8.1	⊢ (𝑎 = 𝐴 → (𝜑 ↔ 𝜓))
br8.2	⊢ (𝑏 = 𝐵 → (𝜓 ↔ 𝜒))
br8.3	⊢ (𝑐 = 𝐶 → (𝜒 ↔ 𝜃))
br8.4	⊢ (𝑑 = 𝐷 → (𝜃 ↔ 𝜏))
br8.5	⊢ (𝑒 = 𝐸 → (𝜏 ↔ 𝜂))
br8.6	⊢ (𝑓 = 𝐹 → (𝜂 ↔ 𝜁))
br8.7	⊢ (𝑔 = 𝐺 → (𝜁 ↔ 𝜎))
br8.8	⊢ (ℎ = 𝐻 → (𝜎 ↔ 𝜌))
br8.9	⊢ (𝑥 = 𝑋 → 𝑃 = 𝑄)
br8.10	⊢ 𝑅 = {⟨𝑝, 𝑞⟩ ∣ ∃𝑥 ∈ 𝑆 ∃𝑎 ∈ 𝑃 ∃𝑏 ∈ 𝑃 ∃𝑐 ∈ 𝑃 ∃𝑑 ∈ 𝑃 ∃𝑒 ∈ 𝑃 ∃𝑓 ∈ 𝑃 ∃𝑔 ∈ 𝑃 ∃ℎ ∈ 𝑃 (𝑝 = ⟨⟨𝑎, 𝑏⟩, ⟨𝑐, 𝑑⟩⟩ ∧ 𝑞 = ⟨⟨𝑒, 𝑓⟩, ⟨𝑔, ℎ⟩⟩ ∧ 𝜑)}

**Assertion**
Ref	Expression
br8	⊢ (((𝑋 ∈ 𝑆 ∧ 𝐴 ∈ 𝑄 ∧ 𝐵 ∈ 𝑄) ∧ (𝐶 ∈ 𝑄 ∧ 𝐷 ∈ 𝑄 ∧ 𝐸 ∈ 𝑄) ∧ (𝐹 ∈ 𝑄 ∧ 𝐺 ∈ 𝑄 ∧ 𝐻 ∈ 𝑄)) → (⟨⟨𝐴, 𝐵⟩, ⟨𝐶, 𝐷⟩⟩𝑅⟨⟨𝐸, 𝐹⟩, ⟨𝐺, 𝐻⟩⟩ ↔ 𝜌))

Distinct variable groups: 𝜒,𝑏 𝜂,𝑒 𝑎,𝑏,𝑐,𝑑,𝑒,𝑓,𝑔,ℎ,𝑝,𝑞,𝑃 𝜓,𝑎 𝜎,𝑔 𝑥,𝑎,𝐴,𝑏,𝑐,𝑑,𝑒,𝑓,𝑔,ℎ,𝑝,𝑞 𝜑,𝑝,𝑞,𝑥 𝜌,𝑎,𝑏,𝑐,𝑑,𝑒,𝑓,𝑔,ℎ,𝑥 𝐵,𝑎,𝑏,𝑐,𝑑,𝑒,𝑓,𝑔,ℎ,𝑝,𝑞,𝑥 𝐶,𝑎,𝑏,𝑐,𝑑,𝑒,𝑓,𝑔,ℎ,𝑝,𝑞,𝑥 𝜏,𝑑 𝐷,𝑎,𝑏,𝑐,𝑑,𝑒,𝑓,𝑔,ℎ,𝑝,𝑞,𝑥 𝐸,𝑎,𝑏,𝑐,𝑑,𝑒,𝑓,𝑔,ℎ,𝑝,𝑞,𝑥 𝜃,𝑐 𝐹,𝑎,𝑏,𝑐,𝑑,𝑒,𝑓,𝑔,ℎ,𝑝,𝑞,𝑥 𝐺,𝑎,𝑏,𝑐,𝑑,𝑒,𝑓,𝑔,ℎ,𝑝,𝑞,𝑥 𝜁,𝑓 𝐻,𝑎,𝑏,𝑐,𝑑,𝑒,𝑓,𝑔,ℎ,𝑝,𝑞,𝑥 𝑆,𝑎,𝑏,𝑐,𝑑,𝑒,𝑓,𝑔,ℎ,𝑝,𝑞,𝑥 𝑄,𝑎,𝑏,𝑐,𝑑,𝑒,𝑓,𝑔,ℎ,𝑥 𝑋,𝑎,𝑏,𝑐,𝑑,𝑒,𝑓,𝑔,ℎ,𝑥 Allowed substitution hints: 𝜑(𝑒, 𝑓, 𝑔, ℎ, 𝑎, 𝑏, 𝑐, 𝑑) 𝜓(𝑥, 𝑒, 𝑓, 𝑔, ℎ, 𝑞, 𝑝, 𝑏, 𝑐, 𝑑) 𝜒(𝑥, 𝑒, 𝑓, 𝑔, ℎ, 𝑞, 𝑝, 𝑎, 𝑐, 𝑑) 𝜃(𝑥, 𝑒, 𝑓, 𝑔, ℎ, 𝑞, 𝑝, 𝑎, 𝑏, 𝑑) 𝜏(𝑥, 𝑒, 𝑓, 𝑔, ℎ, 𝑞, 𝑝, 𝑎, 𝑏, 𝑐) 𝜂(𝑥, 𝑓, 𝑔, ℎ, 𝑞, 𝑝, 𝑎, 𝑏, 𝑐, 𝑑) 𝜁(𝑥, 𝑒, 𝑔, ℎ, 𝑞, 𝑝, 𝑎, 𝑏, 𝑐, 𝑑) 𝜎(𝑥, 𝑒, 𝑓, ℎ, 𝑞, 𝑝, 𝑎, 𝑏, 𝑐, 𝑑) 𝜌(𝑞, 𝑝) 𝑃(𝑥) 𝑄(𝑞, 𝑝) 𝑅(𝑥, 𝑒, 𝑓, 𝑔, ℎ, 𝑞, 𝑝, 𝑎, 𝑏, 𝑐, 𝑑) 𝑋(𝑞, 𝑝)

**Proof of Theorem br8**
Step	Hyp	Ref	Expression
1		opex 5432	. . 3 ⊢ ⟨⟨𝐴, 𝐵⟩, ⟨𝐶, 𝐷⟩⟩ ∈ V
2		opex 5432	. . 3 ⊢ ⟨⟨𝐸, 𝐹⟩, ⟨𝐺, 𝐻⟩⟩ ∈ V
3		eqeq1 2765	. . . . . . . . 9 ⊢ (𝑝 = ⟨⟨𝐴, 𝐵⟩, ⟨𝐶, 𝐷⟩⟩ → (𝑝 = ⟨⟨𝑎, 𝑏⟩, ⟨𝑐, 𝑑⟩⟩ ↔ ⟨⟨𝐴, 𝐵⟩, ⟨𝐶, 𝐷⟩⟩ = ⟨⟨𝑎, 𝑏⟩, ⟨𝑐, 𝑑⟩⟩))
4	3	3anbi1d 1468	. . . . . . . 8 ⊢ (𝑝 = ⟨⟨𝐴, 𝐵⟩, ⟨𝐶, 𝐷⟩⟩ → ((𝑝 = ⟨⟨𝑎, 𝑏⟩, ⟨𝑐, 𝑑⟩⟩ ∧ 𝑞 = ⟨⟨𝑒, 𝑓⟩, ⟨𝑔, ℎ⟩⟩ ∧ 𝜑) ↔ (⟨⟨𝐴, 𝐵⟩, ⟨𝐶, 𝐷⟩⟩ = ⟨⟨𝑎, 𝑏⟩, ⟨𝑐, 𝑑⟩⟩ ∧ 𝑞 = ⟨⟨𝑒, 𝑓⟩, ⟨𝑔, ℎ⟩⟩ ∧ 𝜑)))
5	4	rexbidv 3187	. . . . . . 7 ⊢ (𝑝 = ⟨⟨𝐴, 𝐵⟩, ⟨𝐶, 𝐷⟩⟩ → (∃ℎ ∈ 𝑃 (𝑝 = ⟨⟨𝑎, 𝑏⟩, ⟨𝑐, 𝑑⟩⟩ ∧ 𝑞 = ⟨⟨𝑒, 𝑓⟩, ⟨𝑔, ℎ⟩⟩ ∧ 𝜑) ↔ ∃ℎ ∈ 𝑃 (⟨⟨𝐴, 𝐵⟩, ⟨𝐶, 𝐷⟩⟩ = ⟨⟨𝑎, 𝑏⟩, ⟨𝑐, 𝑑⟩⟩ ∧ 𝑞 = ⟨⟨𝑒, 𝑓⟩, ⟨𝑔, ℎ⟩⟩ ∧ 𝜑)))
6	5	2rexbidv 3228	. . . . . 6 ⊢ (𝑝 = ⟨⟨𝐴, 𝐵⟩, ⟨𝐶, 𝐷⟩⟩ → (∃𝑓 ∈ 𝑃 ∃𝑔 ∈ 𝑃 ∃ℎ ∈ 𝑃 (𝑝 = ⟨⟨𝑎, 𝑏⟩, ⟨𝑐, 𝑑⟩⟩ ∧ 𝑞 = ⟨⟨𝑒, 𝑓⟩, ⟨𝑔, ℎ⟩⟩ ∧ 𝜑) ↔ ∃𝑓 ∈ 𝑃 ∃𝑔 ∈ 𝑃 ∃ℎ ∈ 𝑃 (⟨⟨𝐴, 𝐵⟩, ⟨𝐶, 𝐷⟩⟩ = ⟨⟨𝑎, 𝑏⟩, ⟨𝑐, 𝑑⟩⟩ ∧ 𝑞 = ⟨⟨𝑒, 𝑓⟩, ⟨𝑔, ℎ⟩⟩ ∧ 𝜑)))
7	6	2rexbidv 3228	. . . . 5 ⊢ (𝑝 = ⟨⟨𝐴, 𝐵⟩, ⟨𝐶, 𝐷⟩⟩ → (∃𝑑 ∈ 𝑃 ∃𝑒 ∈ 𝑃 ∃𝑓 ∈ 𝑃 ∃𝑔 ∈ 𝑃 ∃ℎ ∈ 𝑃 (𝑝 = ⟨⟨𝑎, 𝑏⟩, ⟨𝑐, 𝑑⟩⟩ ∧ 𝑞 = ⟨⟨𝑒, 𝑓⟩, ⟨𝑔, ℎ⟩⟩ ∧ 𝜑) ↔ ∃𝑑 ∈ 𝑃 ∃𝑒 ∈ 𝑃 ∃𝑓 ∈ 𝑃 ∃𝑔 ∈ 𝑃 ∃ℎ ∈ 𝑃 (⟨⟨𝐴, 𝐵⟩, ⟨𝐶, 𝐷⟩⟩ = ⟨⟨𝑎, 𝑏⟩, ⟨𝑐, 𝑑⟩⟩ ∧ 𝑞 = ⟨⟨𝑒, 𝑓⟩, ⟨𝑔, ℎ⟩⟩ ∧ 𝜑)))
8	7	2rexbidv 3228	. . . 4 ⊢ (𝑝 = ⟨⟨𝐴, 𝐵⟩, ⟨𝐶, 𝐷⟩⟩ → (∃𝑏 ∈ 𝑃 ∃𝑐 ∈ 𝑃 ∃𝑑 ∈ 𝑃 ∃𝑒 ∈ 𝑃 ∃𝑓 ∈ 𝑃 ∃𝑔 ∈ 𝑃 ∃ℎ ∈ 𝑃 (𝑝 = ⟨⟨𝑎, 𝑏⟩, ⟨𝑐, 𝑑⟩⟩ ∧ 𝑞 = ⟨⟨𝑒, 𝑓⟩, ⟨𝑔, ℎ⟩⟩ ∧ 𝜑) ↔ ∃𝑏 ∈ 𝑃 ∃𝑐 ∈ 𝑃 ∃𝑑 ∈ 𝑃 ∃𝑒 ∈ 𝑃 ∃𝑓 ∈ 𝑃 ∃𝑔 ∈ 𝑃 ∃ℎ ∈ 𝑃 (⟨⟨𝐴, 𝐵⟩, ⟨𝐶, 𝐷⟩⟩ = ⟨⟨𝑎, 𝑏⟩, ⟨𝑐, 𝑑⟩⟩ ∧ 𝑞 = ⟨⟨𝑒, 𝑓⟩, ⟨𝑔, ℎ⟩⟩ ∧ 𝜑)))
9	8	2rexbidv 3228	. . 3 ⊢ (𝑝 = ⟨⟨𝐴, 𝐵⟩, ⟨𝐶, 𝐷⟩⟩ → (∃𝑥 ∈ 𝑆 ∃𝑎 ∈ 𝑃 ∃𝑏 ∈ 𝑃 ∃𝑐 ∈ 𝑃 ∃𝑑 ∈ 𝑃 ∃𝑒 ∈ 𝑃 ∃𝑓 ∈ 𝑃 ∃𝑔 ∈ 𝑃 ∃ℎ ∈ 𝑃 (𝑝 = ⟨⟨𝑎, 𝑏⟩, ⟨𝑐, 𝑑⟩⟩ ∧ 𝑞 = ⟨⟨𝑒, 𝑓⟩, ⟨𝑔, ℎ⟩⟩ ∧ 𝜑) ↔ ∃𝑥 ∈ 𝑆 ∃𝑎 ∈ 𝑃 ∃𝑏 ∈ 𝑃 ∃𝑐 ∈ 𝑃 ∃𝑑 ∈ 𝑃 ∃𝑒 ∈ 𝑃 ∃𝑓 ∈ 𝑃 ∃𝑔 ∈ 𝑃 ∃ℎ ∈ 𝑃 (⟨⟨𝐴, 𝐵⟩, ⟨𝐶, 𝐷⟩⟩ = ⟨⟨𝑎, 𝑏⟩, ⟨𝑐, 𝑑⟩⟩ ∧ 𝑞 = ⟨⟨𝑒, 𝑓⟩, ⟨𝑔, ℎ⟩⟩ ∧ 𝜑)))
10		eqeq1 2765	. . . . . . . . 9 ⊢ (𝑞 = ⟨⟨𝐸, 𝐹⟩, ⟨𝐺, 𝐻⟩⟩ → (𝑞 = ⟨⟨𝑒, 𝑓⟩, ⟨𝑔, ℎ⟩⟩ ↔ ⟨⟨𝐸, 𝐹⟩, ⟨𝐺, 𝐻⟩⟩ = ⟨⟨𝑒, 𝑓⟩, ⟨𝑔, ℎ⟩⟩))
11	10	3anbi2d 1469	. . . . . . . 8 ⊢ (𝑞 = ⟨⟨𝐸, 𝐹⟩, ⟨𝐺, 𝐻⟩⟩ → ((⟨⟨𝐴, 𝐵⟩, ⟨𝐶, 𝐷⟩⟩ = ⟨⟨𝑎, 𝑏⟩, ⟨𝑐, 𝑑⟩⟩ ∧ 𝑞 = ⟨⟨𝑒, 𝑓⟩, ⟨𝑔, ℎ⟩⟩ ∧ 𝜑) ↔ (⟨⟨𝐴, 𝐵⟩, ⟨𝐶, 𝐷⟩⟩ = ⟨⟨𝑎, 𝑏⟩, ⟨𝑐, 𝑑⟩⟩ ∧ ⟨⟨𝐸, 𝐹⟩, ⟨𝐺, 𝐻⟩⟩ = ⟨⟨𝑒, 𝑓⟩, ⟨𝑔, ℎ⟩⟩ ∧ 𝜑)))
12	11	rexbidv 3187	. . . . . . 7 ⊢ (𝑞 = ⟨⟨𝐸, 𝐹⟩, ⟨𝐺, 𝐻⟩⟩ → (∃ℎ ∈ 𝑃 (⟨⟨𝐴, 𝐵⟩, ⟨𝐶, 𝐷⟩⟩ = ⟨⟨𝑎, 𝑏⟩, ⟨𝑐, 𝑑⟩⟩ ∧ 𝑞 = ⟨⟨𝑒, 𝑓⟩, ⟨𝑔, ℎ⟩⟩ ∧ 𝜑) ↔ ∃ℎ ∈ 𝑃 (⟨⟨𝐴, 𝐵⟩, ⟨𝐶, 𝐷⟩⟩ = ⟨⟨𝑎, 𝑏⟩, ⟨𝑐, 𝑑⟩⟩ ∧ ⟨⟨𝐸, 𝐹⟩, ⟨𝐺, 𝐻⟩⟩ = ⟨⟨𝑒, 𝑓⟩, ⟨𝑔, ℎ⟩⟩ ∧ 𝜑)))
13	12	2rexbidv 3228	. . . . . 6 ⊢ (𝑞 = ⟨⟨𝐸, 𝐹⟩, ⟨𝐺, 𝐻⟩⟩ → (∃𝑓 ∈ 𝑃 ∃𝑔 ∈ 𝑃 ∃ℎ ∈ 𝑃 (⟨⟨𝐴, 𝐵⟩, ⟨𝐶, 𝐷⟩⟩ = ⟨⟨𝑎, 𝑏⟩, ⟨𝑐, 𝑑⟩⟩ ∧ 𝑞 = ⟨⟨𝑒, 𝑓⟩, ⟨𝑔, ℎ⟩⟩ ∧ 𝜑) ↔ ∃𝑓 ∈ 𝑃 ∃𝑔 ∈ 𝑃 ∃ℎ ∈ 𝑃 (⟨⟨𝐴, 𝐵⟩, ⟨𝐶, 𝐷⟩⟩ = ⟨⟨𝑎, 𝑏⟩, ⟨𝑐, 𝑑⟩⟩ ∧ ⟨⟨𝐸, 𝐹⟩, ⟨𝐺, 𝐻⟩⟩ = ⟨⟨𝑒, 𝑓⟩, ⟨𝑔, ℎ⟩⟩ ∧ 𝜑)))
14	13	2rexbidv 3228	. . . . 5 ⊢ (𝑞 = ⟨⟨𝐸, 𝐹⟩, ⟨𝐺, 𝐻⟩⟩ → (∃𝑑 ∈ 𝑃 ∃𝑒 ∈ 𝑃 ∃𝑓 ∈ 𝑃 ∃𝑔 ∈ 𝑃 ∃ℎ ∈ 𝑃 (⟨⟨𝐴, 𝐵⟩, ⟨𝐶, 𝐷⟩⟩ = ⟨⟨𝑎, 𝑏⟩, ⟨𝑐, 𝑑⟩⟩ ∧ 𝑞 = ⟨⟨𝑒, 𝑓⟩, ⟨𝑔, ℎ⟩⟩ ∧ 𝜑) ↔ ∃𝑑 ∈ 𝑃 ∃𝑒 ∈ 𝑃 ∃𝑓 ∈ 𝑃 ∃𝑔 ∈ 𝑃 ∃ℎ ∈ 𝑃 (⟨⟨𝐴, 𝐵⟩, ⟨𝐶, 𝐷⟩⟩ = ⟨⟨𝑎, 𝑏⟩, ⟨𝑐, 𝑑⟩⟩ ∧ ⟨⟨𝐸, 𝐹⟩, ⟨𝐺, 𝐻⟩⟩ = ⟨⟨𝑒, 𝑓⟩, ⟨𝑔, ℎ⟩⟩ ∧ 𝜑)))
15	14	2rexbidv 3228	. . . 4 ⊢ (𝑞 = ⟨⟨𝐸, 𝐹⟩, ⟨𝐺, 𝐻⟩⟩ → (∃𝑏 ∈ 𝑃 ∃𝑐 ∈ 𝑃 ∃𝑑 ∈ 𝑃 ∃𝑒 ∈ 𝑃 ∃𝑓 ∈ 𝑃 ∃𝑔 ∈ 𝑃 ∃ℎ ∈ 𝑃 (⟨⟨𝐴, 𝐵⟩, ⟨𝐶, 𝐷⟩⟩ = ⟨⟨𝑎, 𝑏⟩, ⟨𝑐, 𝑑⟩⟩ ∧ 𝑞 = ⟨⟨𝑒, 𝑓⟩, ⟨𝑔, ℎ⟩⟩ ∧ 𝜑) ↔ ∃𝑏 ∈ 𝑃 ∃𝑐 ∈ 𝑃 ∃𝑑 ∈ 𝑃 ∃𝑒 ∈ 𝑃 ∃𝑓 ∈ 𝑃 ∃𝑔 ∈ 𝑃 ∃ℎ ∈ 𝑃 (⟨⟨𝐴, 𝐵⟩, ⟨𝐶, 𝐷⟩⟩ = ⟨⟨𝑎, 𝑏⟩, ⟨𝑐, 𝑑⟩⟩ ∧ ⟨⟨𝐸, 𝐹⟩, ⟨𝐺, 𝐻⟩⟩ = ⟨⟨𝑒, 𝑓⟩, ⟨𝑔, ℎ⟩⟩ ∧ 𝜑)))
16	15	2rexbidv 3228	. . 3 ⊢ (𝑞 = ⟨⟨𝐸, 𝐹⟩, ⟨𝐺, 𝐻⟩⟩ → (∃𝑥 ∈ 𝑆 ∃𝑎 ∈ 𝑃 ∃𝑏 ∈ 𝑃 ∃𝑐 ∈ 𝑃 ∃𝑑 ∈ 𝑃 ∃𝑒 ∈ 𝑃 ∃𝑓 ∈ 𝑃 ∃𝑔 ∈ 𝑃 ∃ℎ ∈ 𝑃 (⟨⟨𝐴, 𝐵⟩, ⟨𝐶, 𝐷⟩⟩ = ⟨⟨𝑎, 𝑏⟩, ⟨𝑐, 𝑑⟩⟩ ∧ 𝑞 = ⟨⟨𝑒, 𝑓⟩, ⟨𝑔, ℎ⟩⟩ ∧ 𝜑) ↔ ∃𝑥 ∈ 𝑆 ∃𝑎 ∈ 𝑃 ∃𝑏 ∈ 𝑃 ∃𝑐 ∈ 𝑃 ∃𝑑 ∈ 𝑃 ∃𝑒 ∈ 𝑃 ∃𝑓 ∈ 𝑃 ∃𝑔 ∈ 𝑃 ∃ℎ ∈ 𝑃 (⟨⟨𝐴, 𝐵⟩, ⟨𝐶, 𝐷⟩⟩ = ⟨⟨𝑎, 𝑏⟩, ⟨𝑐, 𝑑⟩⟩ ∧ ⟨⟨𝐸, 𝐹⟩, ⟨𝐺, 𝐻⟩⟩ = ⟨⟨𝑒, 𝑓⟩, ⟨𝑔, ℎ⟩⟩ ∧ 𝜑)))
17		br8.10	. . 3 ⊢ 𝑅 = {⟨𝑝, 𝑞⟩ ∣ ∃𝑥 ∈ 𝑆 ∃𝑎 ∈ 𝑃 ∃𝑏 ∈ 𝑃 ∃𝑐 ∈ 𝑃 ∃𝑑 ∈ 𝑃 ∃𝑒 ∈ 𝑃 ∃𝑓 ∈ 𝑃 ∃𝑔 ∈ 𝑃 ∃ℎ ∈ 𝑃 (𝑝 = ⟨⟨𝑎, 𝑏⟩, ⟨𝑐, 𝑑⟩⟩ ∧ 𝑞 = ⟨⟨𝑒, 𝑓⟩, ⟨𝑔, ℎ⟩⟩ ∧ 𝜑)}
18	1, 2, 9, 16, 17	brab 5518	. 2 ⊢ (⟨⟨𝐴, 𝐵⟩, ⟨𝐶, 𝐷⟩⟩𝑅⟨⟨𝐸, 𝐹⟩, ⟨𝐺, 𝐻⟩⟩ ↔ ∃𝑥 ∈ 𝑆 ∃𝑎 ∈ 𝑃 ∃𝑏 ∈ 𝑃 ∃𝑐 ∈ 𝑃 ∃𝑑 ∈ 𝑃 ∃𝑒 ∈ 𝑃 ∃𝑓 ∈ 𝑃 ∃𝑔 ∈ 𝑃 ∃ℎ ∈ 𝑃 (⟨⟨𝐴, 𝐵⟩, ⟨𝐶, 𝐷⟩⟩ = ⟨⟨𝑎, 𝑏⟩, ⟨𝑐, 𝑑⟩⟩ ∧ ⟨⟨𝐸, 𝐹⟩, ⟨𝐺, 𝐻⟩⟩ = ⟨⟨𝑒, 𝑓⟩, ⟨𝑔, ℎ⟩⟩ ∧ 𝜑))
19		opex 5432	. . . . . . . . . . . . . 14 ⊢ ⟨𝑎, 𝑏⟩ ∈ V
20		opex 5432	. . . . . . . . . . . . . 14 ⊢ ⟨𝑐, 𝑑⟩ ∈ V
21	19, 20	opth 5445	. . . . . . . . . . . . 13 ⊢ (⟨⟨𝑎, 𝑏⟩, ⟨𝑐, 𝑑⟩⟩ = ⟨⟨𝐴, 𝐵⟩, ⟨𝐶, 𝐷⟩⟩ ↔ (⟨𝑎, 𝑏⟩ = ⟨𝐴, 𝐵⟩ ∧ ⟨𝑐, 𝑑⟩ = ⟨𝐶, 𝐷⟩))
22		vex 3455	. . . . . . . . . . . . . . . 16 ⊢ 𝑎 ∈ V
23		vex 3455	. . . . . . . . . . . . . . . 16 ⊢ 𝑏 ∈ V
24	22, 23	opth 5445	. . . . . . . . . . . . . . 15 ⊢ (⟨𝑎, 𝑏⟩ = ⟨𝐴, 𝐵⟩ ↔ (𝑎 = 𝐴 ∧ 𝑏 = 𝐵))
25		br8.1	. . . . . . . . . . . . . . . 16 ⊢ (𝑎 = 𝐴 → (𝜑 ↔ 𝜓))
26		br8.2	. . . . . . . . . . . . . . . 16 ⊢ (𝑏 = 𝐵 → (𝜓 ↔ 𝜒))
27	25, 26	sylan9bb 519	. . . . . . . . . . . . . . 15 ⊢ ((𝑎 = 𝐴 ∧ 𝑏 = 𝐵) → (𝜑 ↔ 𝜒))
28	24, 27	sylbi 220	. . . . . . . . . . . . . 14 ⊢ (⟨𝑎, 𝑏⟩ = ⟨𝐴, 𝐵⟩ → (𝜑 ↔ 𝜒))
29		vex 3455	. . . . . . . . . . . . . . . 16 ⊢ 𝑐 ∈ V
30		vex 3455	. . . . . . . . . . . . . . . 16 ⊢ 𝑑 ∈ V
31	29, 30	opth 5445	. . . . . . . . . . . . . . 15 ⊢ (⟨𝑐, 𝑑⟩ = ⟨𝐶, 𝐷⟩ ↔ (𝑐 = 𝐶 ∧ 𝑑 = 𝐷))
32		br8.3	. . . . . . . . . . . . . . . 16 ⊢ (𝑐 = 𝐶 → (𝜒 ↔ 𝜃))
33		br8.4	. . . . . . . . . . . . . . . 16 ⊢ (𝑑 = 𝐷 → (𝜃 ↔ 𝜏))
34	32, 33	sylan9bb 519	. . . . . . . . . . . . . . 15 ⊢ ((𝑐 = 𝐶 ∧ 𝑑 = 𝐷) → (𝜒 ↔ 𝜏))
35	31, 34	sylbi 220	. . . . . . . . . . . . . 14 ⊢ (⟨𝑐, 𝑑⟩ = ⟨𝐶, 𝐷⟩ → (𝜒 ↔ 𝜏))
36	28, 35	sylan9bb 519	. . . . . . . . . . . . 13 ⊢ ((⟨𝑎, 𝑏⟩ = ⟨𝐴, 𝐵⟩ ∧ ⟨𝑐, 𝑑⟩ = ⟨𝐶, 𝐷⟩) → (𝜑 ↔ 𝜏))
37	21, 36	sylbi 220	. . . . . . . . . . . 12 ⊢ (⟨⟨𝑎, 𝑏⟩, ⟨𝑐, 𝑑⟩⟩ = ⟨⟨𝐴, 𝐵⟩, ⟨𝐶, 𝐷⟩⟩ → (𝜑 ↔ 𝜏))
38	37	eqcoms 2769	. . . . . . . . . . 11 ⊢ (⟨⟨𝐴, 𝐵⟩, ⟨𝐶, 𝐷⟩⟩ = ⟨⟨𝑎, 𝑏⟩, ⟨𝑐, 𝑑⟩⟩ → (𝜑 ↔ 𝜏))
39		opex 5432	. . . . . . . . . . . . . 14 ⊢ ⟨𝑒, 𝑓⟩ ∈ V
40		opex 5432	. . . . . . . . . . . . . 14 ⊢ ⟨𝑔, ℎ⟩ ∈ V
41	39, 40	opth 5445	. . . . . . . . . . . . 13 ⊢ (⟨⟨𝑒, 𝑓⟩, ⟨𝑔, ℎ⟩⟩ = ⟨⟨𝐸, 𝐹⟩, ⟨𝐺, 𝐻⟩⟩ ↔ (⟨𝑒, 𝑓⟩ = ⟨𝐸, 𝐹⟩ ∧ ⟨𝑔, ℎ⟩ = ⟨𝐺, 𝐻⟩))
42		vex 3455	. . . . . . . . . . . . . . . 16 ⊢ 𝑒 ∈ V
43		vex 3455	. . . . . . . . . . . . . . . 16 ⊢ 𝑓 ∈ V
44	42, 43	opth 5445	. . . . . . . . . . . . . . 15 ⊢ (⟨𝑒, 𝑓⟩ = ⟨𝐸, 𝐹⟩ ↔ (𝑒 = 𝐸 ∧ 𝑓 = 𝐹))
45		br8.5	. . . . . . . . . . . . . . . 16 ⊢ (𝑒 = 𝐸 → (𝜏 ↔ 𝜂))
46		br8.6	. . . . . . . . . . . . . . . 16 ⊢ (𝑓 = 𝐹 → (𝜂 ↔ 𝜁))
47	45, 46	sylan9bb 519	. . . . . . . . . . . . . . 15 ⊢ ((𝑒 = 𝐸 ∧ 𝑓 = 𝐹) → (𝜏 ↔ 𝜁))
48	44, 47	sylbi 220	. . . . . . . . . . . . . 14 ⊢ (⟨𝑒, 𝑓⟩ = ⟨𝐸, 𝐹⟩ → (𝜏 ↔ 𝜁))
49		vex 3455	. . . . . . . . . . . . . . . 16 ⊢ 𝑔 ∈ V
50		vex 3455	. . . . . . . . . . . . . . . 16 ⊢ ℎ ∈ V
51	49, 50	opth 5445	. . . . . . . . . . . . . . 15 ⊢ (⟨𝑔, ℎ⟩ = ⟨𝐺, 𝐻⟩ ↔ (𝑔 = 𝐺 ∧ ℎ = 𝐻))
52		br8.7	. . . . . . . . . . . . . . . 16 ⊢ (𝑔 = 𝐺 → (𝜁 ↔ 𝜎))
53		br8.8	. . . . . . . . . . . . . . . 16 ⊢ (ℎ = 𝐻 → (𝜎 ↔ 𝜌))
54	52, 53	sylan9bb 519	. . . . . . . . . . . . . . 15 ⊢ ((𝑔 = 𝐺 ∧ ℎ = 𝐻) → (𝜁 ↔ 𝜌))
55	51, 54	sylbi 220	. . . . . . . . . . . . . 14 ⊢ (⟨𝑔, ℎ⟩ = ⟨𝐺, 𝐻⟩ → (𝜁 ↔ 𝜌))
56	48, 55	sylan9bb 519	. . . . . . . . . . . . 13 ⊢ ((⟨𝑒, 𝑓⟩ = ⟨𝐸, 𝐹⟩ ∧ ⟨𝑔, ℎ⟩ = ⟨𝐺, 𝐻⟩) → (𝜏 ↔ 𝜌))
57	41, 56	sylbi 220	. . . . . . . . . . . 12 ⊢ (⟨⟨𝑒, 𝑓⟩, ⟨𝑔, ℎ⟩⟩ = ⟨⟨𝐸, 𝐹⟩, ⟨𝐺, 𝐻⟩⟩ → (𝜏 ↔ 𝜌))
58	57	eqcoms 2769	. . . . . . . . . . 11 ⊢ (⟨⟨𝐸, 𝐹⟩, ⟨𝐺, 𝐻⟩⟩ = ⟨⟨𝑒, 𝑓⟩, ⟨𝑔, ℎ⟩⟩ → (𝜏 ↔ 𝜌))
59	38, 58	sylan9bb 519	. . . . . . . . . 10 ⊢ ((⟨⟨𝐴, 𝐵⟩, ⟨𝐶, 𝐷⟩⟩ = ⟨⟨𝑎, 𝑏⟩, ⟨𝑐, 𝑑⟩⟩ ∧ ⟨⟨𝐸, 𝐹⟩, ⟨𝐺, 𝐻⟩⟩ = ⟨⟨𝑒, 𝑓⟩, ⟨𝑔, ℎ⟩⟩) → (𝜑 ↔ 𝜌))
60	59	biimp3a 1498	. . . . . . . . 9 ⊢ ((⟨⟨𝐴, 𝐵⟩, ⟨𝐶, 𝐷⟩⟩ = ⟨⟨𝑎, 𝑏⟩, ⟨𝑐, 𝑑⟩⟩ ∧ ⟨⟨𝐸, 𝐹⟩, ⟨𝐺, 𝐻⟩⟩ = ⟨⟨𝑒, 𝑓⟩, ⟨𝑔, ℎ⟩⟩ ∧ 𝜑) → 𝜌)
61	60	a1i 11	. . . . . . . 8 ⊢ ((((((((𝑋 ∈ 𝑆 ∧ 𝐴 ∈ 𝑄 ∧ 𝐵 ∈ 𝑄) ∧ (𝐶 ∈ 𝑄 ∧ 𝐷 ∈ 𝑄 ∧ 𝐸 ∈ 𝑄) ∧ (𝐹 ∈ 𝑄 ∧ 𝐺 ∈ 𝑄 ∧ 𝐻 ∈ 𝑄)) ∧ (𝑥 ∈ 𝑆 ∧ 𝑎 ∈ 𝑃)) ∧ (𝑏 ∈ 𝑃 ∧ 𝑐 ∈ 𝑃)) ∧ (𝑑 ∈ 𝑃 ∧ 𝑒 ∈ 𝑃)) ∧ (𝑓 ∈ 𝑃 ∧ 𝑔 ∈ 𝑃)) ∧ ℎ ∈ 𝑃) → ((⟨⟨𝐴, 𝐵⟩, ⟨𝐶, 𝐷⟩⟩ = ⟨⟨𝑎, 𝑏⟩, ⟨𝑐, 𝑑⟩⟩ ∧ ⟨⟨𝐸, 𝐹⟩, ⟨𝐺, 𝐻⟩⟩ = ⟨⟨𝑒, 𝑓⟩, ⟨𝑔, ℎ⟩⟩ ∧ 𝜑) → 𝜌))
62	61	rexlimdva 3164	. . . . . . 7 ⊢ (((((((𝑋 ∈ 𝑆 ∧ 𝐴 ∈ 𝑄 ∧ 𝐵 ∈ 𝑄) ∧ (𝐶 ∈ 𝑄 ∧ 𝐷 ∈ 𝑄 ∧ 𝐸 ∈ 𝑄) ∧ (𝐹 ∈ 𝑄 ∧ 𝐺 ∈ 𝑄 ∧ 𝐻 ∈ 𝑄)) ∧ (𝑥 ∈ 𝑆 ∧ 𝑎 ∈ 𝑃)) ∧ (𝑏 ∈ 𝑃 ∧ 𝑐 ∈ 𝑃)) ∧ (𝑑 ∈ 𝑃 ∧ 𝑒 ∈ 𝑃)) ∧ (𝑓 ∈ 𝑃 ∧ 𝑔 ∈ 𝑃)) → (∃ℎ ∈ 𝑃 (⟨⟨𝐴, 𝐵⟩, ⟨𝐶, 𝐷⟩⟩ = ⟨⟨𝑎, 𝑏⟩, ⟨𝑐, 𝑑⟩⟩ ∧ ⟨⟨𝐸, 𝐹⟩, ⟨𝐺, 𝐻⟩⟩ = ⟨⟨𝑒, 𝑓⟩, ⟨𝑔, ℎ⟩⟩ ∧ 𝜑) → 𝜌))
63	62	rexlimdvva 3220	. . . . . 6 ⊢ ((((((𝑋 ∈ 𝑆 ∧ 𝐴 ∈ 𝑄 ∧ 𝐵 ∈ 𝑄) ∧ (𝐶 ∈ 𝑄 ∧ 𝐷 ∈ 𝑄 ∧ 𝐸 ∈ 𝑄) ∧ (𝐹 ∈ 𝑄 ∧ 𝐺 ∈ 𝑄 ∧ 𝐻 ∈ 𝑄)) ∧ (𝑥 ∈ 𝑆 ∧ 𝑎 ∈ 𝑃)) ∧ (𝑏 ∈ 𝑃 ∧ 𝑐 ∈ 𝑃)) ∧ (𝑑 ∈ 𝑃 ∧ 𝑒 ∈ 𝑃)) → (∃𝑓 ∈ 𝑃 ∃𝑔 ∈ 𝑃 ∃ℎ ∈ 𝑃 (⟨⟨𝐴, 𝐵⟩, ⟨𝐶, 𝐷⟩⟩ = ⟨⟨𝑎, 𝑏⟩, ⟨𝑐, 𝑑⟩⟩ ∧ ⟨⟨𝐸, 𝐹⟩, ⟨𝐺, 𝐻⟩⟩ = ⟨⟨𝑒, 𝑓⟩, ⟨𝑔, ℎ⟩⟩ ∧ 𝜑) → 𝜌))
64	63	rexlimdvva 3220	. . . . 5 ⊢ (((((𝑋 ∈ 𝑆 ∧ 𝐴 ∈ 𝑄 ∧ 𝐵 ∈ 𝑄) ∧ (𝐶 ∈ 𝑄 ∧ 𝐷 ∈ 𝑄 ∧ 𝐸 ∈ 𝑄) ∧ (𝐹 ∈ 𝑄 ∧ 𝐺 ∈ 𝑄 ∧ 𝐻 ∈ 𝑄)) ∧ (𝑥 ∈ 𝑆 ∧ 𝑎 ∈ 𝑃)) ∧ (𝑏 ∈ 𝑃 ∧ 𝑐 ∈ 𝑃)) → (∃𝑑 ∈ 𝑃 ∃𝑒 ∈ 𝑃 ∃𝑓 ∈ 𝑃 ∃𝑔 ∈ 𝑃 ∃ℎ ∈ 𝑃 (⟨⟨𝐴, 𝐵⟩, ⟨𝐶, 𝐷⟩⟩ = ⟨⟨𝑎, 𝑏⟩, ⟨𝑐, 𝑑⟩⟩ ∧ ⟨⟨𝐸, 𝐹⟩, ⟨𝐺, 𝐻⟩⟩ = ⟨⟨𝑒, 𝑓⟩, ⟨𝑔, ℎ⟩⟩ ∧ 𝜑) → 𝜌))
65	64	rexlimdvva 3220	. . . 4 ⊢ ((((𝑋 ∈ 𝑆 ∧ 𝐴 ∈ 𝑄 ∧ 𝐵 ∈ 𝑄) ∧ (𝐶 ∈ 𝑄 ∧ 𝐷 ∈ 𝑄 ∧ 𝐸 ∈ 𝑄) ∧ (𝐹 ∈ 𝑄 ∧ 𝐺 ∈ 𝑄 ∧ 𝐻 ∈ 𝑄)) ∧ (𝑥 ∈ 𝑆 ∧ 𝑎 ∈ 𝑃)) → (∃𝑏 ∈ 𝑃 ∃𝑐 ∈ 𝑃 ∃𝑑 ∈ 𝑃 ∃𝑒 ∈ 𝑃 ∃𝑓 ∈ 𝑃 ∃𝑔 ∈ 𝑃 ∃ℎ ∈ 𝑃 (⟨⟨𝐴, 𝐵⟩, ⟨𝐶, 𝐷⟩⟩ = ⟨⟨𝑎, 𝑏⟩, ⟨𝑐, 𝑑⟩⟩ ∧ ⟨⟨𝐸, 𝐹⟩, ⟨𝐺, 𝐻⟩⟩ = ⟨⟨𝑒, 𝑓⟩, ⟨𝑔, ℎ⟩⟩ ∧ 𝜑) → 𝜌))
66	65	rexlimdvva 3220	. . 3 ⊢ (((𝑋 ∈ 𝑆 ∧ 𝐴 ∈ 𝑄 ∧ 𝐵 ∈ 𝑄) ∧ (𝐶 ∈ 𝑄 ∧ 𝐷 ∈ 𝑄 ∧ 𝐸 ∈ 𝑄) ∧ (𝐹 ∈ 𝑄 ∧ 𝐺 ∈ 𝑄 ∧ 𝐻 ∈ 𝑄)) → (∃𝑥 ∈ 𝑆 ∃𝑎 ∈ 𝑃 ∃𝑏 ∈ 𝑃 ∃𝑐 ∈ 𝑃 ∃𝑑 ∈ 𝑃 ∃𝑒 ∈ 𝑃 ∃𝑓 ∈ 𝑃 ∃𝑔 ∈ 𝑃 ∃ℎ ∈ 𝑃 (⟨⟨𝐴, 𝐵⟩, ⟨𝐶, 𝐷⟩⟩ = ⟨⟨𝑎, 𝑏⟩, ⟨𝑐, 𝑑⟩⟩ ∧ ⟨⟨𝐸, 𝐹⟩, ⟨𝐺, 𝐻⟩⟩ = ⟨⟨𝑒, 𝑓⟩, ⟨𝑔, ℎ⟩⟩ ∧ 𝜑) → 𝜌))
67		simpl11 1267	. . . . 5 ⊢ ((((𝑋 ∈ 𝑆 ∧ 𝐴 ∈ 𝑄 ∧ 𝐵 ∈ 𝑄) ∧ (𝐶 ∈ 𝑄 ∧ 𝐷 ∈ 𝑄 ∧ 𝐸 ∈ 𝑄) ∧ (𝐹 ∈ 𝑄 ∧ 𝐺 ∈ 𝑄 ∧ 𝐻 ∈ 𝑄)) ∧ 𝜌) → 𝑋 ∈ 𝑆)
68		simpl12 1268	. . . . . 6 ⊢ ((((𝑋 ∈ 𝑆 ∧ 𝐴 ∈ 𝑄 ∧ 𝐵 ∈ 𝑄) ∧ (𝐶 ∈ 𝑄 ∧ 𝐷 ∈ 𝑄 ∧ 𝐸 ∈ 𝑄) ∧ (𝐹 ∈ 𝑄 ∧ 𝐺 ∈ 𝑄 ∧ 𝐻 ∈ 𝑄)) ∧ 𝜌) → 𝐴 ∈ 𝑄)
69		simpl13 1269	. . . . . 6 ⊢ ((((𝑋 ∈ 𝑆 ∧ 𝐴 ∈ 𝑄 ∧ 𝐵 ∈ 𝑄) ∧ (𝐶 ∈ 𝑄 ∧ 𝐷 ∈ 𝑄 ∧ 𝐸 ∈ 𝑄) ∧ (𝐹 ∈ 𝑄 ∧ 𝐺 ∈ 𝑄 ∧ 𝐻 ∈ 𝑄)) ∧ 𝜌) → 𝐵 ∈ 𝑄)
70		simpl21 1270	. . . . . 6 ⊢ ((((𝑋 ∈ 𝑆 ∧ 𝐴 ∈ 𝑄 ∧ 𝐵 ∈ 𝑄) ∧ (𝐶 ∈ 𝑄 ∧ 𝐷 ∈ 𝑄 ∧ 𝐸 ∈ 𝑄) ∧ (𝐹 ∈ 𝑄 ∧ 𝐺 ∈ 𝑄 ∧ 𝐻 ∈ 𝑄)) ∧ 𝜌) → 𝐶 ∈ 𝑄)
71		simpl22 1271	. . . . . . 7 ⊢ ((((𝑋 ∈ 𝑆 ∧ 𝐴 ∈ 𝑄 ∧ 𝐵 ∈ 𝑄) ∧ (𝐶 ∈ 𝑄 ∧ 𝐷 ∈ 𝑄 ∧ 𝐸 ∈ 𝑄) ∧ (𝐹 ∈ 𝑄 ∧ 𝐺 ∈ 𝑄 ∧ 𝐻 ∈ 𝑄)) ∧ 𝜌) → 𝐷 ∈ 𝑄)
72		simpl23 1272	. . . . . . 7 ⊢ ((((𝑋 ∈ 𝑆 ∧ 𝐴 ∈ 𝑄 ∧ 𝐵 ∈ 𝑄) ∧ (𝐶 ∈ 𝑄 ∧ 𝐷 ∈ 𝑄 ∧ 𝐸 ∈ 𝑄) ∧ (𝐹 ∈ 𝑄 ∧ 𝐺 ∈ 𝑄 ∧ 𝐻 ∈ 𝑄)) ∧ 𝜌) → 𝐸 ∈ 𝑄)
73		simpl31 1273	. . . . . . 7 ⊢ ((((𝑋 ∈ 𝑆 ∧ 𝐴 ∈ 𝑄 ∧ 𝐵 ∈ 𝑄) ∧ (𝐶 ∈ 𝑄 ∧ 𝐷 ∈ 𝑄 ∧ 𝐸 ∈ 𝑄) ∧ (𝐹 ∈ 𝑄 ∧ 𝐺 ∈ 𝑄 ∧ 𝐻 ∈ 𝑄)) ∧ 𝜌) → 𝐹 ∈ 𝑄)
74		simpl32 1274	. . . . . . . 8 ⊢ ((((𝑋 ∈ 𝑆 ∧ 𝐴 ∈ 𝑄 ∧ 𝐵 ∈ 𝑄) ∧ (𝐶 ∈ 𝑄 ∧ 𝐷 ∈ 𝑄 ∧ 𝐸 ∈ 𝑄) ∧ (𝐹 ∈ 𝑄 ∧ 𝐺 ∈ 𝑄 ∧ 𝐻 ∈ 𝑄)) ∧ 𝜌) → 𝐺 ∈ 𝑄)
75		simpl33 1275	. . . . . . . 8 ⊢ ((((𝑋 ∈ 𝑆 ∧ 𝐴 ∈ 𝑄 ∧ 𝐵 ∈ 𝑄) ∧ (𝐶 ∈ 𝑄 ∧ 𝐷 ∈ 𝑄 ∧ 𝐸 ∈ 𝑄) ∧ (𝐹 ∈ 𝑄 ∧ 𝐺 ∈ 𝑄 ∧ 𝐻 ∈ 𝑄)) ∧ 𝜌) → 𝐻 ∈ 𝑄)
76		eqidd 2762	. . . . . . . 8 ⊢ ((((𝑋 ∈ 𝑆 ∧ 𝐴 ∈ 𝑄 ∧ 𝐵 ∈ 𝑄) ∧ (𝐶 ∈ 𝑄 ∧ 𝐷 ∈ 𝑄 ∧ 𝐸 ∈ 𝑄) ∧ (𝐹 ∈ 𝑄 ∧ 𝐺 ∈ 𝑄 ∧ 𝐻 ∈ 𝑄)) ∧ 𝜌) → ⟨⟨𝐴, 𝐵⟩, ⟨𝐶, 𝐷⟩⟩ = ⟨⟨𝐴, 𝐵⟩, ⟨𝐶, 𝐷⟩⟩)
77		eqidd 2762	. . . . . . . 8 ⊢ ((((𝑋 ∈ 𝑆 ∧ 𝐴 ∈ 𝑄 ∧ 𝐵 ∈ 𝑄) ∧ (𝐶 ∈ 𝑄 ∧ 𝐷 ∈ 𝑄 ∧ 𝐸 ∈ 𝑄) ∧ (𝐹 ∈ 𝑄 ∧ 𝐺 ∈ 𝑄 ∧ 𝐻 ∈ 𝑄)) ∧ 𝜌) → ⟨⟨𝐸, 𝐹⟩, ⟨𝐺, 𝐻⟩⟩ = ⟨⟨𝐸, 𝐹⟩, ⟨𝐺, 𝐻⟩⟩)
78		simpr 490	. . . . . . . 8 ⊢ ((((𝑋 ∈ 𝑆 ∧ 𝐴 ∈ 𝑄 ∧ 𝐵 ∈ 𝑄) ∧ (𝐶 ∈ 𝑄 ∧ 𝐷 ∈ 𝑄 ∧ 𝐸 ∈ 𝑄) ∧ (𝐹 ∈ 𝑄 ∧ 𝐺 ∈ 𝑄 ∧ 𝐻 ∈ 𝑄)) ∧ 𝜌) → 𝜌)
79		opeq1 4833	. . . . . . . . . . . 12 ⊢ (𝑔 = 𝐺 → ⟨𝑔, ℎ⟩ = ⟨𝐺, ℎ⟩)
80	79	opeq2d 4840	. . . . . . . . . . 11 ⊢ (𝑔 = 𝐺 → ⟨⟨𝐸, 𝐹⟩, ⟨𝑔, ℎ⟩⟩ = ⟨⟨𝐸, 𝐹⟩, ⟨𝐺, ℎ⟩⟩)
81	80	eqeq2d 2772	. . . . . . . . . 10 ⊢ (𝑔 = 𝐺 → (⟨⟨𝐸, 𝐹⟩, ⟨𝐺, 𝐻⟩⟩ = ⟨⟨𝐸, 𝐹⟩, ⟨𝑔, ℎ⟩⟩ ↔ ⟨⟨𝐸, 𝐹⟩, ⟨𝐺, 𝐻⟩⟩ = ⟨⟨𝐸, 𝐹⟩, ⟨𝐺, ℎ⟩⟩))
82	81, 52	3anbi23d 1467	. . . . . . . . 9 ⊢ (𝑔 = 𝐺 → ((⟨⟨𝐴, 𝐵⟩, ⟨𝐶, 𝐷⟩⟩ = ⟨⟨𝐴, 𝐵⟩, ⟨𝐶, 𝐷⟩⟩ ∧ ⟨⟨𝐸, 𝐹⟩, ⟨𝐺, 𝐻⟩⟩ = ⟨⟨𝐸, 𝐹⟩, ⟨𝑔, ℎ⟩⟩ ∧ 𝜁) ↔ (⟨⟨𝐴, 𝐵⟩, ⟨𝐶, 𝐷⟩⟩ = ⟨⟨𝐴, 𝐵⟩, ⟨𝐶, 𝐷⟩⟩ ∧ ⟨⟨𝐸, 𝐹⟩, ⟨𝐺, 𝐻⟩⟩ = ⟨⟨𝐸, 𝐹⟩, ⟨𝐺, ℎ⟩⟩ ∧ 𝜎)))
83		opeq2 4834	. . . . . . . . . . . 12 ⊢ (ℎ = 𝐻 → ⟨𝐺, ℎ⟩ = ⟨𝐺, 𝐻⟩)
84	83	opeq2d 4840	. . . . . . . . . . 11 ⊢ (ℎ = 𝐻 → ⟨⟨𝐸, 𝐹⟩, ⟨𝐺, ℎ⟩⟩ = ⟨⟨𝐸, 𝐹⟩, ⟨𝐺, 𝐻⟩⟩)
85	84	eqeq2d 2772	. . . . . . . . . 10 ⊢ (ℎ = 𝐻 → (⟨⟨𝐸, 𝐹⟩, ⟨𝐺, 𝐻⟩⟩ = ⟨⟨𝐸, 𝐹⟩, ⟨𝐺, ℎ⟩⟩ ↔ ⟨⟨𝐸, 𝐹⟩, ⟨𝐺, 𝐻⟩⟩ = ⟨⟨𝐸, 𝐹⟩, ⟨𝐺, 𝐻⟩⟩))
86	85, 53	3anbi23d 1467	. . . . . . . . 9 ⊢ (ℎ = 𝐻 → ((⟨⟨𝐴, 𝐵⟩, ⟨𝐶, 𝐷⟩⟩ = ⟨⟨𝐴, 𝐵⟩, ⟨𝐶, 𝐷⟩⟩ ∧ ⟨⟨𝐸, 𝐹⟩, ⟨𝐺, 𝐻⟩⟩ = ⟨⟨𝐸, 𝐹⟩, ⟨𝐺, ℎ⟩⟩ ∧ 𝜎) ↔ (⟨⟨𝐴, 𝐵⟩, ⟨𝐶, 𝐷⟩⟩ = ⟨⟨𝐴, 𝐵⟩, ⟨𝐶, 𝐷⟩⟩ ∧ ⟨⟨𝐸, 𝐹⟩, ⟨𝐺, 𝐻⟩⟩ = ⟨⟨𝐸, 𝐹⟩, ⟨𝐺, 𝐻⟩⟩ ∧ 𝜌)))
87	82, 86	rspc2ev 3589	. . . . . . . 8 ⊢ ((𝐺 ∈ 𝑄 ∧ 𝐻 ∈ 𝑄 ∧ (⟨⟨𝐴, 𝐵⟩, ⟨𝐶, 𝐷⟩⟩ = ⟨⟨𝐴, 𝐵⟩, ⟨𝐶, 𝐷⟩⟩ ∧ ⟨⟨𝐸, 𝐹⟩, ⟨𝐺, 𝐻⟩⟩ = ⟨⟨𝐸, 𝐹⟩, ⟨𝐺, 𝐻⟩⟩ ∧ 𝜌)) → ∃𝑔 ∈ 𝑄 ∃ℎ ∈ 𝑄 (⟨⟨𝐴, 𝐵⟩, ⟨𝐶, 𝐷⟩⟩ = ⟨⟨𝐴, 𝐵⟩, ⟨𝐶, 𝐷⟩⟩ ∧ ⟨⟨𝐸, 𝐹⟩, ⟨𝐺, 𝐻⟩⟩ = ⟨⟨𝐸, 𝐹⟩, ⟨𝑔, ℎ⟩⟩ ∧ 𝜁))
88	74, 75, 76, 77, 78, 87	syl113anc 1409	. . . . . . 7 ⊢ ((((𝑋 ∈ 𝑆 ∧ 𝐴 ∈ 𝑄 ∧ 𝐵 ∈ 𝑄) ∧ (𝐶 ∈ 𝑄 ∧ 𝐷 ∈ 𝑄 ∧ 𝐸 ∈ 𝑄) ∧ (𝐹 ∈ 𝑄 ∧ 𝐺 ∈ 𝑄 ∧ 𝐻 ∈ 𝑄)) ∧ 𝜌) → ∃𝑔 ∈ 𝑄 ∃ℎ ∈ 𝑄 (⟨⟨𝐴, 𝐵⟩, ⟨𝐶, 𝐷⟩⟩ = ⟨⟨𝐴, 𝐵⟩, ⟨𝐶, 𝐷⟩⟩ ∧ ⟨⟨𝐸, 𝐹⟩, ⟨𝐺, 𝐻⟩⟩ = ⟨⟨𝐸, 𝐹⟩, ⟨𝑔, ℎ⟩⟩ ∧ 𝜁))
89		opeq2 4834	. . . . . . . . . . . 12 ⊢ (𝑑 = 𝐷 → ⟨𝐶, 𝑑⟩ = ⟨𝐶, 𝐷⟩)
90	89	opeq2d 4840	. . . . . . . . . . 11 ⊢ (𝑑 = 𝐷 → ⟨⟨𝐴, 𝐵⟩, ⟨𝐶, 𝑑⟩⟩ = ⟨⟨𝐴, 𝐵⟩, ⟨𝐶, 𝐷⟩⟩)
91	90	eqeq2d 2772	. . . . . . . . . 10 ⊢ (𝑑 = 𝐷 → (⟨⟨𝐴, 𝐵⟩, ⟨𝐶, 𝐷⟩⟩ = ⟨⟨𝐴, 𝐵⟩, ⟨𝐶, 𝑑⟩⟩ ↔ ⟨⟨𝐴, 𝐵⟩, ⟨𝐶, 𝐷⟩⟩ = ⟨⟨𝐴, 𝐵⟩, ⟨𝐶, 𝐷⟩⟩))
92	91, 33	3anbi13d 1466	. . . . . . . . 9 ⊢ (𝑑 = 𝐷 → ((⟨⟨𝐴, 𝐵⟩, ⟨𝐶, 𝐷⟩⟩ = ⟨⟨𝐴, 𝐵⟩, ⟨𝐶, 𝑑⟩⟩ ∧ ⟨⟨𝐸, 𝐹⟩, ⟨𝐺, 𝐻⟩⟩ = ⟨⟨𝑒, 𝑓⟩, ⟨𝑔, ℎ⟩⟩ ∧ 𝜃) ↔ (⟨⟨𝐴, 𝐵⟩, ⟨𝐶, 𝐷⟩⟩ = ⟨⟨𝐴, 𝐵⟩, ⟨𝐶, 𝐷⟩⟩ ∧ ⟨⟨𝐸, 𝐹⟩, ⟨𝐺, 𝐻⟩⟩ = ⟨⟨𝑒, 𝑓⟩, ⟨𝑔, ℎ⟩⟩ ∧ 𝜏)))
93	92	2rexbidv 3228	. . . . . . . 8 ⊢ (𝑑 = 𝐷 → (∃𝑔 ∈ 𝑄 ∃ℎ ∈ 𝑄 (⟨⟨𝐴, 𝐵⟩, ⟨𝐶, 𝐷⟩⟩ = ⟨⟨𝐴, 𝐵⟩, ⟨𝐶, 𝑑⟩⟩ ∧ ⟨⟨𝐸, 𝐹⟩, ⟨𝐺, 𝐻⟩⟩ = ⟨⟨𝑒, 𝑓⟩, ⟨𝑔, ℎ⟩⟩ ∧ 𝜃) ↔ ∃𝑔 ∈ 𝑄 ∃ℎ ∈ 𝑄 (⟨⟨𝐴, 𝐵⟩, ⟨𝐶, 𝐷⟩⟩ = ⟨⟨𝐴, 𝐵⟩, ⟨𝐶, 𝐷⟩⟩ ∧ ⟨⟨𝐸, 𝐹⟩, ⟨𝐺, 𝐻⟩⟩ = ⟨⟨𝑒, 𝑓⟩, ⟨𝑔, ℎ⟩⟩ ∧ 𝜏)))
94		opeq1 4833	. . . . . . . . . . . 12 ⊢ (𝑒 = 𝐸 → ⟨𝑒, 𝑓⟩ = ⟨𝐸, 𝑓⟩)
95	94	opeq1d 4839	. . . . . . . . . . 11 ⊢ (𝑒 = 𝐸 → ⟨⟨𝑒, 𝑓⟩, ⟨𝑔, ℎ⟩⟩ = ⟨⟨𝐸, 𝑓⟩, ⟨𝑔, ℎ⟩⟩)
96	95	eqeq2d 2772	. . . . . . . . . 10 ⊢ (𝑒 = 𝐸 → (⟨⟨𝐸, 𝐹⟩, ⟨𝐺, 𝐻⟩⟩ = ⟨⟨𝑒, 𝑓⟩, ⟨𝑔, ℎ⟩⟩ ↔ ⟨⟨𝐸, 𝐹⟩, ⟨𝐺, 𝐻⟩⟩ = ⟨⟨𝐸, 𝑓⟩, ⟨𝑔, ℎ⟩⟩))
97	96, 45	3anbi23d 1467	. . . . . . . . 9 ⊢ (𝑒 = 𝐸 → ((⟨⟨𝐴, 𝐵⟩, ⟨𝐶, 𝐷⟩⟩ = ⟨⟨𝐴, 𝐵⟩, ⟨𝐶, 𝐷⟩⟩ ∧ ⟨⟨𝐸, 𝐹⟩, ⟨𝐺, 𝐻⟩⟩ = ⟨⟨𝑒, 𝑓⟩, ⟨𝑔, ℎ⟩⟩ ∧ 𝜏) ↔ (⟨⟨𝐴, 𝐵⟩, ⟨𝐶, 𝐷⟩⟩ = ⟨⟨𝐴, 𝐵⟩, ⟨𝐶, 𝐷⟩⟩ ∧ ⟨⟨𝐸, 𝐹⟩, ⟨𝐺, 𝐻⟩⟩ = ⟨⟨𝐸, 𝑓⟩, ⟨𝑔, ℎ⟩⟩ ∧ 𝜂)))
98	97	2rexbidv 3228	. . . . . . . 8 ⊢ (𝑒 = 𝐸 → (∃𝑔 ∈ 𝑄 ∃ℎ ∈ 𝑄 (⟨⟨𝐴, 𝐵⟩, ⟨𝐶, 𝐷⟩⟩ = ⟨⟨𝐴, 𝐵⟩, ⟨𝐶, 𝐷⟩⟩ ∧ ⟨⟨𝐸, 𝐹⟩, ⟨𝐺, 𝐻⟩⟩ = ⟨⟨𝑒, 𝑓⟩, ⟨𝑔, ℎ⟩⟩ ∧ 𝜏) ↔ ∃𝑔 ∈ 𝑄 ∃ℎ ∈ 𝑄 (⟨⟨𝐴, 𝐵⟩, ⟨𝐶, 𝐷⟩⟩ = ⟨⟨𝐴, 𝐵⟩, ⟨𝐶, 𝐷⟩⟩ ∧ ⟨⟨𝐸, 𝐹⟩, ⟨𝐺, 𝐻⟩⟩ = ⟨⟨𝐸, 𝑓⟩, ⟨𝑔, ℎ⟩⟩ ∧ 𝜂)))
99		opeq2 4834	. . . . . . . . . . . 12 ⊢ (𝑓 = 𝐹 → ⟨𝐸, 𝑓⟩ = ⟨𝐸, 𝐹⟩)
100	99	opeq1d 4839	. . . . . . . . . . 11 ⊢ (𝑓 = 𝐹 → ⟨⟨𝐸, 𝑓⟩, ⟨𝑔, ℎ⟩⟩ = ⟨⟨𝐸, 𝐹⟩, ⟨𝑔, ℎ⟩⟩)
101	100	eqeq2d 2772	. . . . . . . . . 10 ⊢ (𝑓 = 𝐹 → (⟨⟨𝐸, 𝐹⟩, ⟨𝐺, 𝐻⟩⟩ = ⟨⟨𝐸, 𝑓⟩, ⟨𝑔, ℎ⟩⟩ ↔ ⟨⟨𝐸, 𝐹⟩, ⟨𝐺, 𝐻⟩⟩ = ⟨⟨𝐸, 𝐹⟩, ⟨𝑔, ℎ⟩⟩))
102	101, 46	3anbi23d 1467	. . . . . . . . 9 ⊢ (𝑓 = 𝐹 → ((⟨⟨𝐴, 𝐵⟩, ⟨𝐶, 𝐷⟩⟩ = ⟨⟨𝐴, 𝐵⟩, ⟨𝐶, 𝐷⟩⟩ ∧ ⟨⟨𝐸, 𝐹⟩, ⟨𝐺, 𝐻⟩⟩ = ⟨⟨𝐸, 𝑓⟩, ⟨𝑔, ℎ⟩⟩ ∧ 𝜂) ↔ (⟨⟨𝐴, 𝐵⟩, ⟨𝐶, 𝐷⟩⟩ = ⟨⟨𝐴, 𝐵⟩, ⟨𝐶, 𝐷⟩⟩ ∧ ⟨⟨𝐸, 𝐹⟩, ⟨𝐺, 𝐻⟩⟩ = ⟨⟨𝐸, 𝐹⟩, ⟨𝑔, ℎ⟩⟩ ∧ 𝜁)))
103	102	2rexbidv 3228	. . . . . . . 8 ⊢ (𝑓 = 𝐹 → (∃𝑔 ∈ 𝑄 ∃ℎ ∈ 𝑄 (⟨⟨𝐴, 𝐵⟩, ⟨𝐶, 𝐷⟩⟩ = ⟨⟨𝐴, 𝐵⟩, ⟨𝐶, 𝐷⟩⟩ ∧ ⟨⟨𝐸, 𝐹⟩, ⟨𝐺, 𝐻⟩⟩ = ⟨⟨𝐸, 𝑓⟩, ⟨𝑔, ℎ⟩⟩ ∧ 𝜂) ↔ ∃𝑔 ∈ 𝑄 ∃ℎ ∈ 𝑄 (⟨⟨𝐴, 𝐵⟩, ⟨𝐶, 𝐷⟩⟩ = ⟨⟨𝐴, 𝐵⟩, ⟨𝐶, 𝐷⟩⟩ ∧ ⟨⟨𝐸, 𝐹⟩, ⟨𝐺, 𝐻⟩⟩ = ⟨⟨𝐸, 𝐹⟩, ⟨𝑔, ℎ⟩⟩ ∧ 𝜁)))
104	93, 98, 103	rspc3ev 3593	. . . . . . 7 ⊢ (((𝐷 ∈ 𝑄 ∧ 𝐸 ∈ 𝑄 ∧ 𝐹 ∈ 𝑄) ∧ ∃𝑔 ∈ 𝑄 ∃ℎ ∈ 𝑄 (⟨⟨𝐴, 𝐵⟩, ⟨𝐶, 𝐷⟩⟩ = ⟨⟨𝐴, 𝐵⟩, ⟨𝐶, 𝐷⟩⟩ ∧ ⟨⟨𝐸, 𝐹⟩, ⟨𝐺, 𝐻⟩⟩ = ⟨⟨𝐸, 𝐹⟩, ⟨𝑔, ℎ⟩⟩ ∧ 𝜁)) → ∃𝑑 ∈ 𝑄 ∃𝑒 ∈ 𝑄 ∃𝑓 ∈ 𝑄 ∃𝑔 ∈ 𝑄 ∃ℎ ∈ 𝑄 (⟨⟨𝐴, 𝐵⟩, ⟨𝐶, 𝐷⟩⟩ = ⟨⟨𝐴, 𝐵⟩, ⟨𝐶, 𝑑⟩⟩ ∧ ⟨⟨𝐸, 𝐹⟩, ⟨𝐺, 𝐻⟩⟩ = ⟨⟨𝑒, 𝑓⟩, ⟨𝑔, ℎ⟩⟩ ∧ 𝜃))
105	71, 72, 73, 88, 104	syl31anc 1400	. . . . . 6 ⊢ ((((𝑋 ∈ 𝑆 ∧ 𝐴 ∈ 𝑄 ∧ 𝐵 ∈ 𝑄) ∧ (𝐶 ∈ 𝑄 ∧ 𝐷 ∈ 𝑄 ∧ 𝐸 ∈ 𝑄) ∧ (𝐹 ∈ 𝑄 ∧ 𝐺 ∈ 𝑄 ∧ 𝐻 ∈ 𝑄)) ∧ 𝜌) → ∃𝑑 ∈ 𝑄 ∃𝑒 ∈ 𝑄 ∃𝑓 ∈ 𝑄 ∃𝑔 ∈ 𝑄 ∃ℎ ∈ 𝑄 (⟨⟨𝐴, 𝐵⟩, ⟨𝐶, 𝐷⟩⟩ = ⟨⟨𝐴, 𝐵⟩, ⟨𝐶, 𝑑⟩⟩ ∧ ⟨⟨𝐸, 𝐹⟩, ⟨𝐺, 𝐻⟩⟩ = ⟨⟨𝑒, 𝑓⟩, ⟨𝑔, ℎ⟩⟩ ∧ 𝜃))
106		opeq1 4833	. . . . . . . . . . . . 13 ⊢ (𝑎 = 𝐴 → ⟨𝑎, 𝑏⟩ = ⟨𝐴, 𝑏⟩)
107	106	opeq1d 4839	. . . . . . . . . . . 12 ⊢ (𝑎 = 𝐴 → ⟨⟨𝑎, 𝑏⟩, ⟨𝑐, 𝑑⟩⟩ = ⟨⟨𝐴, 𝑏⟩, ⟨𝑐, 𝑑⟩⟩)
108	107	eqeq2d 2772	. . . . . . . . . . 11 ⊢ (𝑎 = 𝐴 → (⟨⟨𝐴, 𝐵⟩, ⟨𝐶, 𝐷⟩⟩ = ⟨⟨𝑎, 𝑏⟩, ⟨𝑐, 𝑑⟩⟩ ↔ ⟨⟨𝐴, 𝐵⟩, ⟨𝐶, 𝐷⟩⟩ = ⟨⟨𝐴, 𝑏⟩, ⟨𝑐, 𝑑⟩⟩))
109	108, 25	3anbi13d 1466	. . . . . . . . . 10 ⊢ (𝑎 = 𝐴 → ((⟨⟨𝐴, 𝐵⟩, ⟨𝐶, 𝐷⟩⟩ = ⟨⟨𝑎, 𝑏⟩, ⟨𝑐, 𝑑⟩⟩ ∧ ⟨⟨𝐸, 𝐹⟩, ⟨𝐺, 𝐻⟩⟩ = ⟨⟨𝑒, 𝑓⟩, ⟨𝑔, ℎ⟩⟩ ∧ 𝜑) ↔ (⟨⟨𝐴, 𝐵⟩, ⟨𝐶, 𝐷⟩⟩ = ⟨⟨𝐴, 𝑏⟩, ⟨𝑐, 𝑑⟩⟩ ∧ ⟨⟨𝐸, 𝐹⟩, ⟨𝐺, 𝐻⟩⟩ = ⟨⟨𝑒, 𝑓⟩, ⟨𝑔, ℎ⟩⟩ ∧ 𝜓)))
110	109	rexbidv 3187	. . . . . . . . 9 ⊢ (𝑎 = 𝐴 → (∃ℎ ∈ 𝑄 (⟨⟨𝐴, 𝐵⟩, ⟨𝐶, 𝐷⟩⟩ = ⟨⟨𝑎, 𝑏⟩, ⟨𝑐, 𝑑⟩⟩ ∧ ⟨⟨𝐸, 𝐹⟩, ⟨𝐺, 𝐻⟩⟩ = ⟨⟨𝑒, 𝑓⟩, ⟨𝑔, ℎ⟩⟩ ∧ 𝜑) ↔ ∃ℎ ∈ 𝑄 (⟨⟨𝐴, 𝐵⟩, ⟨𝐶, 𝐷⟩⟩ = ⟨⟨𝐴, 𝑏⟩, ⟨𝑐, 𝑑⟩⟩ ∧ ⟨⟨𝐸, 𝐹⟩, ⟨𝐺, 𝐻⟩⟩ = ⟨⟨𝑒, 𝑓⟩, ⟨𝑔, ℎ⟩⟩ ∧ 𝜓)))
111	110	2rexbidv 3228	. . . . . . . 8 ⊢ (𝑎 = 𝐴 → (∃𝑓 ∈ 𝑄 ∃𝑔 ∈ 𝑄 ∃ℎ ∈ 𝑄 (⟨⟨𝐴, 𝐵⟩, ⟨𝐶, 𝐷⟩⟩ = ⟨⟨𝑎, 𝑏⟩, ⟨𝑐, 𝑑⟩⟩ ∧ ⟨⟨𝐸, 𝐹⟩, ⟨𝐺, 𝐻⟩⟩ = ⟨⟨𝑒, 𝑓⟩, ⟨𝑔, ℎ⟩⟩ ∧ 𝜑) ↔ ∃𝑓 ∈ 𝑄 ∃𝑔 ∈ 𝑄 ∃ℎ ∈ 𝑄 (⟨⟨𝐴, 𝐵⟩, ⟨𝐶, 𝐷⟩⟩ = ⟨⟨𝐴, 𝑏⟩, ⟨𝑐, 𝑑⟩⟩ ∧ ⟨⟨𝐸, 𝐹⟩, ⟨𝐺, 𝐻⟩⟩ = ⟨⟨𝑒, 𝑓⟩, ⟨𝑔, ℎ⟩⟩ ∧ 𝜓)))
112	111	2rexbidv 3228	. . . . . . 7 ⊢ (𝑎 = 𝐴 → (∃𝑑 ∈ 𝑄 ∃𝑒 ∈ 𝑄 ∃𝑓 ∈ 𝑄 ∃𝑔 ∈ 𝑄 ∃ℎ ∈ 𝑄 (⟨⟨𝐴, 𝐵⟩, ⟨𝐶, 𝐷⟩⟩ = ⟨⟨𝑎, 𝑏⟩, ⟨𝑐, 𝑑⟩⟩ ∧ ⟨⟨𝐸, 𝐹⟩, ⟨𝐺, 𝐻⟩⟩ = ⟨⟨𝑒, 𝑓⟩, ⟨𝑔, ℎ⟩⟩ ∧ 𝜑) ↔ ∃𝑑 ∈ 𝑄 ∃𝑒 ∈ 𝑄 ∃𝑓 ∈ 𝑄 ∃𝑔 ∈ 𝑄 ∃ℎ ∈ 𝑄 (⟨⟨𝐴, 𝐵⟩, ⟨𝐶, 𝐷⟩⟩ = ⟨⟨𝐴, 𝑏⟩, ⟨𝑐, 𝑑⟩⟩ ∧ ⟨⟨𝐸, 𝐹⟩, ⟨𝐺, 𝐻⟩⟩ = ⟨⟨𝑒, 𝑓⟩, ⟨𝑔, ℎ⟩⟩ ∧ 𝜓)))
113		opeq2 4834	. . . . . . . . . . . . 13 ⊢ (𝑏 = 𝐵 → ⟨𝐴, 𝑏⟩ = ⟨𝐴, 𝐵⟩)
114	113	opeq1d 4839	. . . . . . . . . . . 12 ⊢ (𝑏 = 𝐵 → ⟨⟨𝐴, 𝑏⟩, ⟨𝑐, 𝑑⟩⟩ = ⟨⟨𝐴, 𝐵⟩, ⟨𝑐, 𝑑⟩⟩)
115	114	eqeq2d 2772	. . . . . . . . . . 11 ⊢ (𝑏 = 𝐵 → (⟨⟨𝐴, 𝐵⟩, ⟨𝐶, 𝐷⟩⟩ = ⟨⟨𝐴, 𝑏⟩, ⟨𝑐, 𝑑⟩⟩ ↔ ⟨⟨𝐴, 𝐵⟩, ⟨𝐶, 𝐷⟩⟩ = ⟨⟨𝐴, 𝐵⟩, ⟨𝑐, 𝑑⟩⟩))
116	115, 26	3anbi13d 1466	. . . . . . . . . 10 ⊢ (𝑏 = 𝐵 → ((⟨⟨𝐴, 𝐵⟩, ⟨𝐶, 𝐷⟩⟩ = ⟨⟨𝐴, 𝑏⟩, ⟨𝑐, 𝑑⟩⟩ ∧ ⟨⟨𝐸, 𝐹⟩, ⟨𝐺, 𝐻⟩⟩ = ⟨⟨𝑒, 𝑓⟩, ⟨𝑔, ℎ⟩⟩ ∧ 𝜓) ↔ (⟨⟨𝐴, 𝐵⟩, ⟨𝐶, 𝐷⟩⟩ = ⟨⟨𝐴, 𝐵⟩, ⟨𝑐, 𝑑⟩⟩ ∧ ⟨⟨𝐸, 𝐹⟩, ⟨𝐺, 𝐻⟩⟩ = ⟨⟨𝑒, 𝑓⟩, ⟨𝑔, ℎ⟩⟩ ∧ 𝜒)))
117	116	rexbidv 3187	. . . . . . . . 9 ⊢ (𝑏 = 𝐵 → (∃ℎ ∈ 𝑄 (⟨⟨𝐴, 𝐵⟩, ⟨𝐶, 𝐷⟩⟩ = ⟨⟨𝐴, 𝑏⟩, ⟨𝑐, 𝑑⟩⟩ ∧ ⟨⟨𝐸, 𝐹⟩, ⟨𝐺, 𝐻⟩⟩ = ⟨⟨𝑒, 𝑓⟩, ⟨𝑔, ℎ⟩⟩ ∧ 𝜓) ↔ ∃ℎ ∈ 𝑄 (⟨⟨𝐴, 𝐵⟩, ⟨𝐶, 𝐷⟩⟩ = ⟨⟨𝐴, 𝐵⟩, ⟨𝑐, 𝑑⟩⟩ ∧ ⟨⟨𝐸, 𝐹⟩, ⟨𝐺, 𝐻⟩⟩ = ⟨⟨𝑒, 𝑓⟩, ⟨𝑔, ℎ⟩⟩ ∧ 𝜒)))
118	117	2rexbidv 3228	. . . . . . . 8 ⊢ (𝑏 = 𝐵 → (∃𝑓 ∈ 𝑄 ∃𝑔 ∈ 𝑄 ∃ℎ ∈ 𝑄 (⟨⟨𝐴, 𝐵⟩, ⟨𝐶, 𝐷⟩⟩ = ⟨⟨𝐴, 𝑏⟩, ⟨𝑐, 𝑑⟩⟩ ∧ ⟨⟨𝐸, 𝐹⟩, ⟨𝐺, 𝐻⟩⟩ = ⟨⟨𝑒, 𝑓⟩, ⟨𝑔, ℎ⟩⟩ ∧ 𝜓) ↔ ∃𝑓 ∈ 𝑄 ∃𝑔 ∈ 𝑄 ∃ℎ ∈ 𝑄 (⟨⟨𝐴, 𝐵⟩, ⟨𝐶, 𝐷⟩⟩ = ⟨⟨𝐴, 𝐵⟩, ⟨𝑐, 𝑑⟩⟩ ∧ ⟨⟨𝐸, 𝐹⟩, ⟨𝐺, 𝐻⟩⟩ = ⟨⟨𝑒, 𝑓⟩, ⟨𝑔, ℎ⟩⟩ ∧ 𝜒)))
119	118	2rexbidv 3228	. . . . . . 7 ⊢ (𝑏 = 𝐵 → (∃𝑑 ∈ 𝑄 ∃𝑒 ∈ 𝑄 ∃𝑓 ∈ 𝑄 ∃𝑔 ∈ 𝑄 ∃ℎ ∈ 𝑄 (⟨⟨𝐴, 𝐵⟩, ⟨𝐶, 𝐷⟩⟩ = ⟨⟨𝐴, 𝑏⟩, ⟨𝑐, 𝑑⟩⟩ ∧ ⟨⟨𝐸, 𝐹⟩, ⟨𝐺, 𝐻⟩⟩ = ⟨⟨𝑒, 𝑓⟩, ⟨𝑔, ℎ⟩⟩ ∧ 𝜓) ↔ ∃𝑑 ∈ 𝑄 ∃𝑒 ∈ 𝑄 ∃𝑓 ∈ 𝑄 ∃𝑔 ∈ 𝑄 ∃ℎ ∈ 𝑄 (⟨⟨𝐴, 𝐵⟩, ⟨𝐶, 𝐷⟩⟩ = ⟨⟨𝐴, 𝐵⟩, ⟨𝑐, 𝑑⟩⟩ ∧ ⟨⟨𝐸, 𝐹⟩, ⟨𝐺, 𝐻⟩⟩ = ⟨⟨𝑒, 𝑓⟩, ⟨𝑔, ℎ⟩⟩ ∧ 𝜒)))
120		opeq1 4833	. . . . . . . . . . . . 13 ⊢ (𝑐 = 𝐶 → ⟨𝑐, 𝑑⟩ = ⟨𝐶, 𝑑⟩)
121	120	opeq2d 4840	. . . . . . . . . . . 12 ⊢ (𝑐 = 𝐶 → ⟨⟨𝐴, 𝐵⟩, ⟨𝑐, 𝑑⟩⟩ = ⟨⟨𝐴, 𝐵⟩, ⟨𝐶, 𝑑⟩⟩)
122	121	eqeq2d 2772	. . . . . . . . . . 11 ⊢ (𝑐 = 𝐶 → (⟨⟨𝐴, 𝐵⟩, ⟨𝐶, 𝐷⟩⟩ = ⟨⟨𝐴, 𝐵⟩, ⟨𝑐, 𝑑⟩⟩ ↔ ⟨⟨𝐴, 𝐵⟩, ⟨𝐶, 𝐷⟩⟩ = ⟨⟨𝐴, 𝐵⟩, ⟨𝐶, 𝑑⟩⟩))
123	122, 32	3anbi13d 1466	. . . . . . . . . 10 ⊢ (𝑐 = 𝐶 → ((⟨⟨𝐴, 𝐵⟩, ⟨𝐶, 𝐷⟩⟩ = ⟨⟨𝐴, 𝐵⟩, ⟨𝑐, 𝑑⟩⟩ ∧ ⟨⟨𝐸, 𝐹⟩, ⟨𝐺, 𝐻⟩⟩ = ⟨⟨𝑒, 𝑓⟩, ⟨𝑔, ℎ⟩⟩ ∧ 𝜒) ↔ (⟨⟨𝐴, 𝐵⟩, ⟨𝐶, 𝐷⟩⟩ = ⟨⟨𝐴, 𝐵⟩, ⟨𝐶, 𝑑⟩⟩ ∧ ⟨⟨𝐸, 𝐹⟩, ⟨𝐺, 𝐻⟩⟩ = ⟨⟨𝑒, 𝑓⟩, ⟨𝑔, ℎ⟩⟩ ∧ 𝜃)))
124	123	rexbidv 3187	. . . . . . . . 9 ⊢ (𝑐 = 𝐶 → (∃ℎ ∈ 𝑄 (⟨⟨𝐴, 𝐵⟩, ⟨𝐶, 𝐷⟩⟩ = ⟨⟨𝐴, 𝐵⟩, ⟨𝑐, 𝑑⟩⟩ ∧ ⟨⟨𝐸, 𝐹⟩, ⟨𝐺, 𝐻⟩⟩ = ⟨⟨𝑒, 𝑓⟩, ⟨𝑔, ℎ⟩⟩ ∧ 𝜒) ↔ ∃ℎ ∈ 𝑄 (⟨⟨𝐴, 𝐵⟩, ⟨𝐶, 𝐷⟩⟩ = ⟨⟨𝐴, 𝐵⟩, ⟨𝐶, 𝑑⟩⟩ ∧ ⟨⟨𝐸, 𝐹⟩, ⟨𝐺, 𝐻⟩⟩ = ⟨⟨𝑒, 𝑓⟩, ⟨𝑔, ℎ⟩⟩ ∧ 𝜃)))
125	124	2rexbidv 3228	. . . . . . . 8 ⊢ (𝑐 = 𝐶 → (∃𝑓 ∈ 𝑄 ∃𝑔 ∈ 𝑄 ∃ℎ ∈ 𝑄 (⟨⟨𝐴, 𝐵⟩, ⟨𝐶, 𝐷⟩⟩ = ⟨⟨𝐴, 𝐵⟩, ⟨𝑐, 𝑑⟩⟩ ∧ ⟨⟨𝐸, 𝐹⟩, ⟨𝐺, 𝐻⟩⟩ = ⟨⟨𝑒, 𝑓⟩, ⟨𝑔, ℎ⟩⟩ ∧ 𝜒) ↔ ∃𝑓 ∈ 𝑄 ∃𝑔 ∈ 𝑄 ∃ℎ ∈ 𝑄 (⟨⟨𝐴, 𝐵⟩, ⟨𝐶, 𝐷⟩⟩ = ⟨⟨𝐴, 𝐵⟩, ⟨𝐶, 𝑑⟩⟩ ∧ ⟨⟨𝐸, 𝐹⟩, ⟨𝐺, 𝐻⟩⟩ = ⟨⟨𝑒, 𝑓⟩, ⟨𝑔, ℎ⟩⟩ ∧ 𝜃)))
126	125	2rexbidv 3228	. . . . . . 7 ⊢ (𝑐 = 𝐶 → (∃𝑑 ∈ 𝑄 ∃𝑒 ∈ 𝑄 ∃𝑓 ∈ 𝑄 ∃𝑔 ∈ 𝑄 ∃ℎ ∈ 𝑄 (⟨⟨𝐴, 𝐵⟩, ⟨𝐶, 𝐷⟩⟩ = ⟨⟨𝐴, 𝐵⟩, ⟨𝑐, 𝑑⟩⟩ ∧ ⟨⟨𝐸, 𝐹⟩, ⟨𝐺, 𝐻⟩⟩ = ⟨⟨𝑒, 𝑓⟩, ⟨𝑔, ℎ⟩⟩ ∧ 𝜒) ↔ ∃𝑑 ∈ 𝑄 ∃𝑒 ∈ 𝑄 ∃𝑓 ∈ 𝑄 ∃𝑔 ∈ 𝑄 ∃ℎ ∈ 𝑄 (⟨⟨𝐴, 𝐵⟩, ⟨𝐶, 𝐷⟩⟩ = ⟨⟨𝐴, 𝐵⟩, ⟨𝐶, 𝑑⟩⟩ ∧ ⟨⟨𝐸, 𝐹⟩, ⟨𝐺, 𝐻⟩⟩ = ⟨⟨𝑒, 𝑓⟩, ⟨𝑔, ℎ⟩⟩ ∧ 𝜃)))
127	112, 119, 126	rspc3ev 3593	. . . . . 6 ⊢ (((𝐴 ∈ 𝑄 ∧ 𝐵 ∈ 𝑄 ∧ 𝐶 ∈ 𝑄) ∧ ∃𝑑 ∈ 𝑄 ∃𝑒 ∈ 𝑄 ∃𝑓 ∈ 𝑄 ∃𝑔 ∈ 𝑄 ∃ℎ ∈ 𝑄 (⟨⟨𝐴, 𝐵⟩, ⟨𝐶, 𝐷⟩⟩ = ⟨⟨𝐴, 𝐵⟩, ⟨𝐶, 𝑑⟩⟩ ∧ ⟨⟨𝐸, 𝐹⟩, ⟨𝐺, 𝐻⟩⟩ = ⟨⟨𝑒, 𝑓⟩, ⟨𝑔, ℎ⟩⟩ ∧ 𝜃)) → ∃𝑎 ∈ 𝑄 ∃𝑏 ∈ 𝑄 ∃𝑐 ∈ 𝑄 ∃𝑑 ∈ 𝑄 ∃𝑒 ∈ 𝑄 ∃𝑓 ∈ 𝑄 ∃𝑔 ∈ 𝑄 ∃ℎ ∈ 𝑄 (⟨⟨𝐴, 𝐵⟩, ⟨𝐶, 𝐷⟩⟩ = ⟨⟨𝑎, 𝑏⟩, ⟨𝑐, 𝑑⟩⟩ ∧ ⟨⟨𝐸, 𝐹⟩, ⟨𝐺, 𝐻⟩⟩ = ⟨⟨𝑒, 𝑓⟩, ⟨𝑔, ℎ⟩⟩ ∧ 𝜑))
128	68, 69, 70, 105, 127	syl31anc 1400	. . . . 5 ⊢ ((((𝑋 ∈ 𝑆 ∧ 𝐴 ∈ 𝑄 ∧ 𝐵 ∈ 𝑄) ∧ (𝐶 ∈ 𝑄 ∧ 𝐷 ∈ 𝑄 ∧ 𝐸 ∈ 𝑄) ∧ (𝐹 ∈ 𝑄 ∧ 𝐺 ∈ 𝑄 ∧ 𝐻 ∈ 𝑄)) ∧ 𝜌) → ∃𝑎 ∈ 𝑄 ∃𝑏 ∈ 𝑄 ∃𝑐 ∈ 𝑄 ∃𝑑 ∈ 𝑄 ∃𝑒 ∈ 𝑄 ∃𝑓 ∈ 𝑄 ∃𝑔 ∈ 𝑄 ∃ℎ ∈ 𝑄 (⟨⟨𝐴, 𝐵⟩, ⟨𝐶, 𝐷⟩⟩ = ⟨⟨𝑎, 𝑏⟩, ⟨𝑐, 𝑑⟩⟩ ∧ ⟨⟨𝐸, 𝐹⟩, ⟨𝐺, 𝐻⟩⟩ = ⟨⟨𝑒, 𝑓⟩, ⟨𝑔, ℎ⟩⟩ ∧ 𝜑))
129		br8.9	. . . . . . 7 ⊢ (𝑥 = 𝑋 → 𝑃 = 𝑄)
130	129	rexeqdv 3321	. . . . . . . . . . . . 13 ⊢ (𝑥 = 𝑋 → (∃ℎ ∈ 𝑃 (⟨⟨𝐴, 𝐵⟩, ⟨𝐶, 𝐷⟩⟩ = ⟨⟨𝑎, 𝑏⟩, ⟨𝑐, 𝑑⟩⟩ ∧ ⟨⟨𝐸, 𝐹⟩, ⟨𝐺, 𝐻⟩⟩ = ⟨⟨𝑒, 𝑓⟩, ⟨𝑔, ℎ⟩⟩ ∧ 𝜑) ↔ ∃ℎ ∈ 𝑄 (⟨⟨𝐴, 𝐵⟩, ⟨𝐶, 𝐷⟩⟩ = ⟨⟨𝑎, 𝑏⟩, ⟨𝑐, 𝑑⟩⟩ ∧ ⟨⟨𝐸, 𝐹⟩, ⟨𝐺, 𝐻⟩⟩ = ⟨⟨𝑒, 𝑓⟩, ⟨𝑔, ℎ⟩⟩ ∧ 𝜑)))
131	129, 130	rexeqbidv 3336	. . . . . . . . . . . 12 ⊢ (𝑥 = 𝑋 → (∃𝑔 ∈ 𝑃 ∃ℎ ∈ 𝑃 (⟨⟨𝐴, 𝐵⟩, ⟨𝐶, 𝐷⟩⟩ = ⟨⟨𝑎, 𝑏⟩, ⟨𝑐, 𝑑⟩⟩ ∧ ⟨⟨𝐸, 𝐹⟩, ⟨𝐺, 𝐻⟩⟩ = ⟨⟨𝑒, 𝑓⟩, ⟨𝑔, ℎ⟩⟩ ∧ 𝜑) ↔ ∃𝑔 ∈ 𝑄 ∃ℎ ∈ 𝑄 (⟨⟨𝐴, 𝐵⟩, ⟨𝐶, 𝐷⟩⟩ = ⟨⟨𝑎, 𝑏⟩, ⟨𝑐, 𝑑⟩⟩ ∧ ⟨⟨𝐸, 𝐹⟩, ⟨𝐺, 𝐻⟩⟩ = ⟨⟨𝑒, 𝑓⟩, ⟨𝑔, ℎ⟩⟩ ∧ 𝜑)))
132	129, 131	rexeqbidv 3336	. . . . . . . . . . 11 ⊢ (𝑥 = 𝑋 → (∃𝑓 ∈ 𝑃 ∃𝑔 ∈ 𝑃 ∃ℎ ∈ 𝑃 (⟨⟨𝐴, 𝐵⟩, ⟨𝐶, 𝐷⟩⟩ = ⟨⟨𝑎, 𝑏⟩, ⟨𝑐, 𝑑⟩⟩ ∧ ⟨⟨𝐸, 𝐹⟩, ⟨𝐺, 𝐻⟩⟩ = ⟨⟨𝑒, 𝑓⟩, ⟨𝑔, ℎ⟩⟩ ∧ 𝜑) ↔ ∃𝑓 ∈ 𝑄 ∃𝑔 ∈ 𝑄 ∃ℎ ∈ 𝑄 (⟨⟨𝐴, 𝐵⟩, ⟨𝐶, 𝐷⟩⟩ = ⟨⟨𝑎, 𝑏⟩, ⟨𝑐, 𝑑⟩⟩ ∧ ⟨⟨𝐸, 𝐹⟩, ⟨𝐺, 𝐻⟩⟩ = ⟨⟨𝑒, 𝑓⟩, ⟨𝑔, ℎ⟩⟩ ∧ 𝜑)))
133	129, 132	rexeqbidv 3336	. . . . . . . . . 10 ⊢ (𝑥 = 𝑋 → (∃𝑒 ∈ 𝑃 ∃𝑓 ∈ 𝑃 ∃𝑔 ∈ 𝑃 ∃ℎ ∈ 𝑃 (⟨⟨𝐴, 𝐵⟩, ⟨𝐶, 𝐷⟩⟩ = ⟨⟨𝑎, 𝑏⟩, ⟨𝑐, 𝑑⟩⟩ ∧ ⟨⟨𝐸, 𝐹⟩, ⟨𝐺, 𝐻⟩⟩ = ⟨⟨𝑒, 𝑓⟩, ⟨𝑔, ℎ⟩⟩ ∧ 𝜑) ↔ ∃𝑒 ∈ 𝑄 ∃𝑓 ∈ 𝑄 ∃𝑔 ∈ 𝑄 ∃ℎ ∈ 𝑄 (⟨⟨𝐴, 𝐵⟩, ⟨𝐶, 𝐷⟩⟩ = ⟨⟨𝑎, 𝑏⟩, ⟨𝑐, 𝑑⟩⟩ ∧ ⟨⟨𝐸, 𝐹⟩, ⟨𝐺, 𝐻⟩⟩ = ⟨⟨𝑒, 𝑓⟩, ⟨𝑔, ℎ⟩⟩ ∧ 𝜑)))
134	129, 133	rexeqbidv 3336	. . . . . . . . 9 ⊢ (𝑥 = 𝑋 → (∃𝑑 ∈ 𝑃 ∃𝑒 ∈ 𝑃 ∃𝑓 ∈ 𝑃 ∃𝑔 ∈ 𝑃 ∃ℎ ∈ 𝑃 (⟨⟨𝐴, 𝐵⟩, ⟨𝐶, 𝐷⟩⟩ = ⟨⟨𝑎, 𝑏⟩, ⟨𝑐, 𝑑⟩⟩ ∧ ⟨⟨𝐸, 𝐹⟩, ⟨𝐺, 𝐻⟩⟩ = ⟨⟨𝑒, 𝑓⟩, ⟨𝑔, ℎ⟩⟩ ∧ 𝜑) ↔ ∃𝑑 ∈ 𝑄 ∃𝑒 ∈ 𝑄 ∃𝑓 ∈ 𝑄 ∃𝑔 ∈ 𝑄 ∃ℎ ∈ 𝑄 (⟨⟨𝐴, 𝐵⟩, ⟨𝐶, 𝐷⟩⟩ = ⟨⟨𝑎, 𝑏⟩, ⟨𝑐, 𝑑⟩⟩ ∧ ⟨⟨𝐸, 𝐹⟩, ⟨𝐺, 𝐻⟩⟩ = ⟨⟨𝑒, 𝑓⟩, ⟨𝑔, ℎ⟩⟩ ∧ 𝜑)))
135	129, 134	rexeqbidv 3336	. . . . . . . 8 ⊢ (𝑥 = 𝑋 → (∃𝑐 ∈ 𝑃 ∃𝑑 ∈ 𝑃 ∃𝑒 ∈ 𝑃 ∃𝑓 ∈ 𝑃 ∃𝑔 ∈ 𝑃 ∃ℎ ∈ 𝑃 (⟨⟨𝐴, 𝐵⟩, ⟨𝐶, 𝐷⟩⟩ = ⟨⟨𝑎, 𝑏⟩, ⟨𝑐, 𝑑⟩⟩ ∧ ⟨⟨𝐸, 𝐹⟩, ⟨𝐺, 𝐻⟩⟩ = ⟨⟨𝑒, 𝑓⟩, ⟨𝑔, ℎ⟩⟩ ∧ 𝜑) ↔ ∃𝑐 ∈ 𝑄 ∃𝑑 ∈ 𝑄 ∃𝑒 ∈ 𝑄 ∃𝑓 ∈ 𝑄 ∃𝑔 ∈ 𝑄 ∃ℎ ∈ 𝑄 (⟨⟨𝐴, 𝐵⟩, ⟨𝐶, 𝐷⟩⟩ = ⟨⟨𝑎, 𝑏⟩, ⟨𝑐, 𝑑⟩⟩ ∧ ⟨⟨𝐸, 𝐹⟩, ⟨𝐺, 𝐻⟩⟩ = ⟨⟨𝑒, 𝑓⟩, ⟨𝑔, ℎ⟩⟩ ∧ 𝜑)))
136	129, 135	rexeqbidv 3336	. . . . . . 7 ⊢ (𝑥 = 𝑋 → (∃𝑏 ∈ 𝑃 ∃𝑐 ∈ 𝑃 ∃𝑑 ∈ 𝑃 ∃𝑒 ∈ 𝑃 ∃𝑓 ∈ 𝑃 ∃𝑔 ∈ 𝑃 ∃ℎ ∈ 𝑃 (⟨⟨𝐴, 𝐵⟩, ⟨𝐶, 𝐷⟩⟩ = ⟨⟨𝑎, 𝑏⟩, ⟨𝑐, 𝑑⟩⟩ ∧ ⟨⟨𝐸, 𝐹⟩, ⟨𝐺, 𝐻⟩⟩ = ⟨⟨𝑒, 𝑓⟩, ⟨𝑔, ℎ⟩⟩ ∧ 𝜑) ↔ ∃𝑏 ∈ 𝑄 ∃𝑐 ∈ 𝑄 ∃𝑑 ∈ 𝑄 ∃𝑒 ∈ 𝑄 ∃𝑓 ∈ 𝑄 ∃𝑔 ∈ 𝑄 ∃ℎ ∈ 𝑄 (⟨⟨𝐴, 𝐵⟩, ⟨𝐶, 𝐷⟩⟩ = ⟨⟨𝑎, 𝑏⟩, ⟨𝑐, 𝑑⟩⟩ ∧ ⟨⟨𝐸, 𝐹⟩, ⟨𝐺, 𝐻⟩⟩ = ⟨⟨𝑒, 𝑓⟩, ⟨𝑔, ℎ⟩⟩ ∧ 𝜑)))
137	129, 136	rexeqbidv 3336	. . . . . 6 ⊢ (𝑥 = 𝑋 → (∃𝑎 ∈ 𝑃 ∃𝑏 ∈ 𝑃 ∃𝑐 ∈ 𝑃 ∃𝑑 ∈ 𝑃 ∃𝑒 ∈ 𝑃 ∃𝑓 ∈ 𝑃 ∃𝑔 ∈ 𝑃 ∃ℎ ∈ 𝑃 (⟨⟨𝐴, 𝐵⟩, ⟨𝐶, 𝐷⟩⟩ = ⟨⟨𝑎, 𝑏⟩, ⟨𝑐, 𝑑⟩⟩ ∧ ⟨⟨𝐸, 𝐹⟩, ⟨𝐺, 𝐻⟩⟩ = ⟨⟨𝑒, 𝑓⟩, ⟨𝑔, ℎ⟩⟩ ∧ 𝜑) ↔ ∃𝑎 ∈ 𝑄 ∃𝑏 ∈ 𝑄 ∃𝑐 ∈ 𝑄 ∃𝑑 ∈ 𝑄 ∃𝑒 ∈ 𝑄 ∃𝑓 ∈ 𝑄 ∃𝑔 ∈ 𝑄 ∃ℎ ∈ 𝑄 (⟨⟨𝐴, 𝐵⟩, ⟨𝐶, 𝐷⟩⟩ = ⟨⟨𝑎, 𝑏⟩, ⟨𝑐, 𝑑⟩⟩ ∧ ⟨⟨𝐸, 𝐹⟩, ⟨𝐺, 𝐻⟩⟩ = ⟨⟨𝑒, 𝑓⟩, ⟨𝑔, ℎ⟩⟩ ∧ 𝜑)))
138	137	rspcev 3577	. . . . 5 ⊢ ((𝑋 ∈ 𝑆 ∧ ∃𝑎 ∈ 𝑄 ∃𝑏 ∈ 𝑄 ∃𝑐 ∈ 𝑄 ∃𝑑 ∈ 𝑄 ∃𝑒 ∈ 𝑄 ∃𝑓 ∈ 𝑄 ∃𝑔 ∈ 𝑄 ∃ℎ ∈ 𝑄 (⟨⟨𝐴, 𝐵⟩, ⟨𝐶, 𝐷⟩⟩ = ⟨⟨𝑎, 𝑏⟩, ⟨𝑐, 𝑑⟩⟩ ∧ ⟨⟨𝐸, 𝐹⟩, ⟨𝐺, 𝐻⟩⟩ = ⟨⟨𝑒, 𝑓⟩, ⟨𝑔, ℎ⟩⟩ ∧ 𝜑)) → ∃𝑥 ∈ 𝑆 ∃𝑎 ∈ 𝑃 ∃𝑏 ∈ 𝑃 ∃𝑐 ∈ 𝑃 ∃𝑑 ∈ 𝑃 ∃𝑒 ∈ 𝑃 ∃𝑓 ∈ 𝑃 ∃𝑔 ∈ 𝑃 ∃ℎ ∈ 𝑃 (⟨⟨𝐴, 𝐵⟩, ⟨𝐶, 𝐷⟩⟩ = ⟨⟨𝑎, 𝑏⟩, ⟨𝑐, 𝑑⟩⟩ ∧ ⟨⟨𝐸, 𝐹⟩, ⟨𝐺, 𝐻⟩⟩ = ⟨⟨𝑒, 𝑓⟩, ⟨𝑔, ℎ⟩⟩ ∧ 𝜑))
139	67, 128, 138	syl2anc 596	. . . 4 ⊢ ((((𝑋 ∈ 𝑆 ∧ 𝐴 ∈ 𝑄 ∧ 𝐵 ∈ 𝑄) ∧ (𝐶 ∈ 𝑄 ∧ 𝐷 ∈ 𝑄 ∧ 𝐸 ∈ 𝑄) ∧ (𝐹 ∈ 𝑄 ∧ 𝐺 ∈ 𝑄 ∧ 𝐻 ∈ 𝑄)) ∧ 𝜌) → ∃𝑥 ∈ 𝑆 ∃𝑎 ∈ 𝑃 ∃𝑏 ∈ 𝑃 ∃𝑐 ∈ 𝑃 ∃𝑑 ∈ 𝑃 ∃𝑒 ∈ 𝑃 ∃𝑓 ∈ 𝑃 ∃𝑔 ∈ 𝑃 ∃ℎ ∈ 𝑃 (⟨⟨𝐴, 𝐵⟩, ⟨𝐶, 𝐷⟩⟩ = ⟨⟨𝑎, 𝑏⟩, ⟨𝑐, 𝑑⟩⟩ ∧ ⟨⟨𝐸, 𝐹⟩, ⟨𝐺, 𝐻⟩⟩ = ⟨⟨𝑒, 𝑓⟩, ⟨𝑔, ℎ⟩⟩ ∧ 𝜑))
140	139	ex 418	. . 3 ⊢ (((𝑋 ∈ 𝑆 ∧ 𝐴 ∈ 𝑄 ∧ 𝐵 ∈ 𝑄) ∧ (𝐶 ∈ 𝑄 ∧ 𝐷 ∈ 𝑄 ∧ 𝐸 ∈ 𝑄) ∧ (𝐹 ∈ 𝑄 ∧ 𝐺 ∈ 𝑄 ∧ 𝐻 ∈ 𝑄)) → (𝜌 → ∃𝑥 ∈ 𝑆 ∃𝑎 ∈ 𝑃 ∃𝑏 ∈ 𝑃 ∃𝑐 ∈ 𝑃 ∃𝑑 ∈ 𝑃 ∃𝑒 ∈ 𝑃 ∃𝑓 ∈ 𝑃 ∃𝑔 ∈ 𝑃 ∃ℎ ∈ 𝑃 (⟨⟨𝐴, 𝐵⟩, ⟨𝐶, 𝐷⟩⟩ = ⟨⟨𝑎, 𝑏⟩, ⟨𝑐, 𝑑⟩⟩ ∧ ⟨⟨𝐸, 𝐹⟩, ⟨𝐺, 𝐻⟩⟩ = ⟨⟨𝑒, 𝑓⟩, ⟨𝑔, ℎ⟩⟩ ∧ 𝜑)))
141	66, 140	impbid 215	. 2 ⊢ (((𝑋 ∈ 𝑆 ∧ 𝐴 ∈ 𝑄 ∧ 𝐵 ∈ 𝑄) ∧ (𝐶 ∈ 𝑄 ∧ 𝐷 ∈ 𝑄 ∧ 𝐸 ∈ 𝑄) ∧ (𝐹 ∈ 𝑄 ∧ 𝐺 ∈ 𝑄 ∧ 𝐻 ∈ 𝑄)) → (∃𝑥 ∈ 𝑆 ∃𝑎 ∈ 𝑃 ∃𝑏 ∈ 𝑃 ∃𝑐 ∈ 𝑃 ∃𝑑 ∈ 𝑃 ∃𝑒 ∈ 𝑃 ∃𝑓 ∈ 𝑃 ∃𝑔 ∈ 𝑃 ∃ℎ ∈ 𝑃 (⟨⟨𝐴, 𝐵⟩, ⟨𝐶, 𝐷⟩⟩ = ⟨⟨𝑎, 𝑏⟩, ⟨𝑐, 𝑑⟩⟩ ∧ ⟨⟨𝐸, 𝐹⟩, ⟨𝐺, 𝐻⟩⟩ = ⟨⟨𝑒, 𝑓⟩, ⟨𝑔, ℎ⟩⟩ ∧ 𝜑) ↔ 𝜌))
142	18, 141	bitrid 286	1 ⊢ (((𝑋 ∈ 𝑆 ∧ 𝐴 ∈ 𝑄 ∧ 𝐵 ∈ 𝑄) ∧ (𝐶 ∈ 𝑄 ∧ 𝐷 ∈ 𝑄 ∧ 𝐸 ∈ 𝑄) ∧ (𝐹 ∈ 𝑄 ∧ 𝐺 ∈ 𝑄 ∧ 𝐻 ∈ 𝑄)) → (⟨⟨𝐴, 𝐵⟩, ⟨𝐶, 𝐷⟩⟩𝑅⟨⟨𝐸, 𝐹⟩, ⟨𝐺, 𝐻⟩⟩ ↔ 𝜌))

Colors of variables: wff setvar class

This proof depends on syntax axioms: → wi 4 ↔ wb 209 ∧ wa 401 ∧ w3a 1103 = wceq 1570 ∈ wcel 2145 ∃wrex 3087 ⟨cop 4590 class class class wbr 5103 {copab 5167

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1828 ax-4 1842 ax-5 1943 ax-6 2000 ax-7 2041 ax-8 2147 ax-9 2155 ax-ext 2733 ax-sep 5249 ax-pr 5391

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3an 1105 df-tru 1573 df-fal 1583 df-ex 1813 df-sb 2100 df-clab 2740 df-cleq 2753 df-clel 2836 df-ral 3078 df-rex 3088 df-rab 3414 df-v 3453 df-dif 3902 df-un 3904 df-in 3906 df-ss 3916 df-nul 4280 df-if 4483 df-sn 4585 df-pr 4587 df-op 4591 df-br 5104 df-opab 5168

This theorem is used by: brofs 36750 brifs 36788 brfs 36824

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