Theorem genpassu 7333

Description: Associativity of upper cuts. Lemma for genpassg 7334. (Contributed by Jim Kingdon, 11-Dec-2019.)

Hypotheses

Ref	Expression
genpelvl.1	⊢ 𝐹 = (𝑤 ∈ P, 𝑣 ∈ P ↦ ⟨{𝑥 ∈ Q ∣ ∃𝑦 ∈ Q ∃𝑧 ∈ Q (𝑦 ∈ (1st ‘𝑤) ∧ 𝑧 ∈ (1st ‘𝑣) ∧ 𝑥 = (𝑦𝐺𝑧))}, {𝑥 ∈ Q ∣ ∃𝑦 ∈ Q ∃𝑧 ∈ Q (𝑦 ∈ (2nd ‘𝑤) ∧ 𝑧 ∈ (2nd ‘𝑣) ∧ 𝑥 = (𝑦𝐺𝑧))}⟩)
genpelvl.2	⊢ ((𝑦 ∈ Q ∧ 𝑧 ∈ Q) → (𝑦𝐺𝑧) ∈ Q)
genpassg.4	⊢ dom 𝐹 = (P × P)
genpassg.5	⊢ ((𝑓 ∈ P ∧ 𝑔 ∈ P) → (𝑓𝐹𝑔) ∈ P)
genpassg.6	⊢ ((𝑓 ∈ Q ∧ 𝑔 ∈ Q ∧ ℎ ∈ Q) → ((𝑓𝐺𝑔)𝐺ℎ) = (𝑓𝐺(𝑔𝐺ℎ)))

Assertion

Ref	Expression
genpassu	⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P ∧ 𝐶 ∈ P) → (2nd ‘((𝐴𝐹𝐵)𝐹𝐶)) = (2nd ‘(𝐴𝐹(𝐵𝐹𝐶))))

Distinct variable groups:   𝑥,𝑦,𝑧,𝑓,𝑔,ℎ,𝑤,𝑣,𝐴   𝑥,𝐵,𝑦,𝑧,𝑓,𝑔,ℎ,𝑤,𝑣   𝑥,𝐺,𝑦,𝑧,𝑓,𝑔,ℎ,𝑤,𝑣   𝑓,𝐹,𝑔   𝐶,𝑓,𝑔,ℎ,𝑣,𝑤,𝑥,𝑦,𝑧   ℎ,𝐹,𝑣,𝑤,𝑥,𝑦,𝑧

Proof of Theorem genpassu

Dummy variable 𝑡 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		prop 7283	. . . . . . . . 9 ⊢ (𝐴 ∈ P → ⟨(1st ‘𝐴), (2nd ‘𝐴)⟩ ∈ P)
2		elprnqu 7290	. . . . . . . . 9 ⊢ ((⟨(1st ‘𝐴), (2nd ‘𝐴)⟩ ∈ P ∧ 𝑓 ∈ (2nd ‘𝐴)) → 𝑓 ∈ Q)
3	1, 2	sylan 281	. . . . . . . 8 ⊢ ((𝐴 ∈ P ∧ 𝑓 ∈ (2nd ‘𝐴)) → 𝑓 ∈ Q)
4		prop 7283	. . . . . . . . . . . . . . . 16 ⊢ (𝐵 ∈ P → ⟨(1st ‘𝐵), (2nd ‘𝐵)⟩ ∈ P)
5		elprnqu 7290	. . . . . . . . . . . . . . . 16 ⊢ ((⟨(1st ‘𝐵), (2nd ‘𝐵)⟩ ∈ P ∧ 𝑔 ∈ (2nd ‘𝐵)) → 𝑔 ∈ Q)
6	4, 5	sylan 281	. . . . . . . . . . . . . . 15 ⊢ ((𝐵 ∈ P ∧ 𝑔 ∈ (2nd ‘𝐵)) → 𝑔 ∈ Q)
7		r19.41v 2587	. . . . . . . . . . . . . . . . 17 ⊢ (∃ℎ ∈ (2nd ‘𝐶)(𝑡 = (𝑔𝐺ℎ) ∧ 𝑥 = (𝑓𝐺𝑡)) ↔ (∃ℎ ∈ (2nd ‘𝐶)𝑡 = (𝑔𝐺ℎ) ∧ 𝑥 = (𝑓𝐺𝑡)))
8		prop 7283	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝐶 ∈ P → ⟨(1st ‘𝐶), (2nd ‘𝐶)⟩ ∈ P)
9		elprnqu 7290	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((⟨(1st ‘𝐶), (2nd ‘𝐶)⟩ ∈ P ∧ ℎ ∈ (2nd ‘𝐶)) → ℎ ∈ Q)
10	8, 9	sylan 281	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝐶 ∈ P ∧ ℎ ∈ (2nd ‘𝐶)) → ℎ ∈ Q)
11		oveq2 5782	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝑡 = (𝑔𝐺ℎ) → (𝑓𝐺𝑡) = (𝑓𝐺(𝑔𝐺ℎ)))
12	11	adantr 274	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝑡 = (𝑔𝐺ℎ) ∧ (𝑓 ∈ Q ∧ 𝑔 ∈ Q ∧ ℎ ∈ Q)) → (𝑓𝐺𝑡) = (𝑓𝐺(𝑔𝐺ℎ)))
13		genpassg.6	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((𝑓 ∈ Q ∧ 𝑔 ∈ Q ∧ ℎ ∈ Q) → ((𝑓𝐺𝑔)𝐺ℎ) = (𝑓𝐺(𝑔𝐺ℎ)))
14	13	adantl 275	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝑡 = (𝑔𝐺ℎ) ∧ (𝑓 ∈ Q ∧ 𝑔 ∈ Q ∧ ℎ ∈ Q)) → ((𝑓𝐺𝑔)𝐺ℎ) = (𝑓𝐺(𝑔𝐺ℎ)))
15	12, 14	eqtr4d 2175	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝑡 = (𝑔𝐺ℎ) ∧ (𝑓 ∈ Q ∧ 𝑔 ∈ Q ∧ ℎ ∈ Q)) → (𝑓𝐺𝑡) = ((𝑓𝐺𝑔)𝐺ℎ))
16	15	eqeq2d 2151	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝑡 = (𝑔𝐺ℎ) ∧ (𝑓 ∈ Q ∧ 𝑔 ∈ Q ∧ ℎ ∈ Q)) → (𝑥 = (𝑓𝐺𝑡) ↔ 𝑥 = ((𝑓𝐺𝑔)𝐺ℎ)))
17	16	expcom 115	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝑓 ∈ Q ∧ 𝑔 ∈ Q ∧ ℎ ∈ Q) → (𝑡 = (𝑔𝐺ℎ) → (𝑥 = (𝑓𝐺𝑡) ↔ 𝑥 = ((𝑓𝐺𝑔)𝐺ℎ))))
18	17	pm5.32d 445	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑓 ∈ Q ∧ 𝑔 ∈ Q ∧ ℎ ∈ Q) → ((𝑡 = (𝑔𝐺ℎ) ∧ 𝑥 = (𝑓𝐺𝑡)) ↔ (𝑡 = (𝑔𝐺ℎ) ∧ 𝑥 = ((𝑓𝐺𝑔)𝐺ℎ))))
19	18	3expa 1181	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝑓 ∈ Q ∧ 𝑔 ∈ Q) ∧ ℎ ∈ Q) → ((𝑡 = (𝑔𝐺ℎ) ∧ 𝑥 = (𝑓𝐺𝑡)) ↔ (𝑡 = (𝑔𝐺ℎ) ∧ 𝑥 = ((𝑓𝐺𝑔)𝐺ℎ))))
20	10, 19	sylan2 284	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝑓 ∈ Q ∧ 𝑔 ∈ Q) ∧ (𝐶 ∈ P ∧ ℎ ∈ (2nd ‘𝐶))) → ((𝑡 = (𝑔𝐺ℎ) ∧ 𝑥 = (𝑓𝐺𝑡)) ↔ (𝑡 = (𝑔𝐺ℎ) ∧ 𝑥 = ((𝑓𝐺𝑔)𝐺ℎ))))
21	20	anassrs 397	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝑓 ∈ Q ∧ 𝑔 ∈ Q) ∧ 𝐶 ∈ P) ∧ ℎ ∈ (2nd ‘𝐶)) → ((𝑡 = (𝑔𝐺ℎ) ∧ 𝑥 = (𝑓𝐺𝑡)) ↔ (𝑡 = (𝑔𝐺ℎ) ∧ 𝑥 = ((𝑓𝐺𝑔)𝐺ℎ))))
22	21	rexbidva 2434	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑓 ∈ Q ∧ 𝑔 ∈ Q) ∧ 𝐶 ∈ P) → (∃ℎ ∈ (2nd ‘𝐶)(𝑡 = (𝑔𝐺ℎ) ∧ 𝑥 = (𝑓𝐺𝑡)) ↔ ∃ℎ ∈ (2nd ‘𝐶)(𝑡 = (𝑔𝐺ℎ) ∧ 𝑥 = ((𝑓𝐺𝑔)𝐺ℎ))))
23	7, 22	syl5rbbr 194	. . . . . . . . . . . . . . . 16 ⊢ (((𝑓 ∈ Q ∧ 𝑔 ∈ Q) ∧ 𝐶 ∈ P) → (∃ℎ ∈ (2nd ‘𝐶)(𝑡 = (𝑔𝐺ℎ) ∧ 𝑥 = ((𝑓𝐺𝑔)𝐺ℎ)) ↔ (∃ℎ ∈ (2nd ‘𝐶)𝑡 = (𝑔𝐺ℎ) ∧ 𝑥 = (𝑓𝐺𝑡))))
24	23	an32s 557	. . . . . . . . . . . . . . 15 ⊢ (((𝑓 ∈ Q ∧ 𝐶 ∈ P) ∧ 𝑔 ∈ Q) → (∃ℎ ∈ (2nd ‘𝐶)(𝑡 = (𝑔𝐺ℎ) ∧ 𝑥 = ((𝑓𝐺𝑔)𝐺ℎ)) ↔ (∃ℎ ∈ (2nd ‘𝐶)𝑡 = (𝑔𝐺ℎ) ∧ 𝑥 = (𝑓𝐺𝑡))))
25	6, 24	sylan2 284	. . . . . . . . . . . . . 14 ⊢ (((𝑓 ∈ Q ∧ 𝐶 ∈ P) ∧ (𝐵 ∈ P ∧ 𝑔 ∈ (2nd ‘𝐵))) → (∃ℎ ∈ (2nd ‘𝐶)(𝑡 = (𝑔𝐺ℎ) ∧ 𝑥 = ((𝑓𝐺𝑔)𝐺ℎ)) ↔ (∃ℎ ∈ (2nd ‘𝐶)𝑡 = (𝑔𝐺ℎ) ∧ 𝑥 = (𝑓𝐺𝑡))))
26	25	anassrs 397	. . . . . . . . . . . . 13 ⊢ ((((𝑓 ∈ Q ∧ 𝐶 ∈ P) ∧ 𝐵 ∈ P) ∧ 𝑔 ∈ (2nd ‘𝐵)) → (∃ℎ ∈ (2nd ‘𝐶)(𝑡 = (𝑔𝐺ℎ) ∧ 𝑥 = ((𝑓𝐺𝑔)𝐺ℎ)) ↔ (∃ℎ ∈ (2nd ‘𝐶)𝑡 = (𝑔𝐺ℎ) ∧ 𝑥 = (𝑓𝐺𝑡))))
27	26	rexbidva 2434	. . . . . . . . . . . 12 ⊢ (((𝑓 ∈ Q ∧ 𝐶 ∈ P) ∧ 𝐵 ∈ P) → (∃𝑔 ∈ (2nd ‘𝐵)∃ℎ ∈ (2nd ‘𝐶)(𝑡 = (𝑔𝐺ℎ) ∧ 𝑥 = ((𝑓𝐺𝑔)𝐺ℎ)) ↔ ∃𝑔 ∈ (2nd ‘𝐵)(∃ℎ ∈ (2nd ‘𝐶)𝑡 = (𝑔𝐺ℎ) ∧ 𝑥 = (𝑓𝐺𝑡))))
28		r19.41v 2587	. . . . . . . . . . . 12 ⊢ (∃𝑔 ∈ (2nd ‘𝐵)(∃ℎ ∈ (2nd ‘𝐶)𝑡 = (𝑔𝐺ℎ) ∧ 𝑥 = (𝑓𝐺𝑡)) ↔ (∃𝑔 ∈ (2nd ‘𝐵)∃ℎ ∈ (2nd ‘𝐶)𝑡 = (𝑔𝐺ℎ) ∧ 𝑥 = (𝑓𝐺𝑡)))
29	27, 28	syl6bb 195	. . . . . . . . . . 11 ⊢ (((𝑓 ∈ Q ∧ 𝐶 ∈ P) ∧ 𝐵 ∈ P) → (∃𝑔 ∈ (2nd ‘𝐵)∃ℎ ∈ (2nd ‘𝐶)(𝑡 = (𝑔𝐺ℎ) ∧ 𝑥 = ((𝑓𝐺𝑔)𝐺ℎ)) ↔ (∃𝑔 ∈ (2nd ‘𝐵)∃ℎ ∈ (2nd ‘𝐶)𝑡 = (𝑔𝐺ℎ) ∧ 𝑥 = (𝑓𝐺𝑡))))
30	29	an31s 559	. . . . . . . . . 10 ⊢ (((𝐵 ∈ P ∧ 𝐶 ∈ P) ∧ 𝑓 ∈ Q) → (∃𝑔 ∈ (2nd ‘𝐵)∃ℎ ∈ (2nd ‘𝐶)(𝑡 = (𝑔𝐺ℎ) ∧ 𝑥 = ((𝑓𝐺𝑔)𝐺ℎ)) ↔ (∃𝑔 ∈ (2nd ‘𝐵)∃ℎ ∈ (2nd ‘𝐶)𝑡 = (𝑔𝐺ℎ) ∧ 𝑥 = (𝑓𝐺𝑡))))
31	30	exbidv 1797	. . . . . . . . 9 ⊢ (((𝐵 ∈ P ∧ 𝐶 ∈ P) ∧ 𝑓 ∈ Q) → (∃𝑡∃𝑔 ∈ (2nd ‘𝐵)∃ℎ ∈ (2nd ‘𝐶)(𝑡 = (𝑔𝐺ℎ) ∧ 𝑥 = ((𝑓𝐺𝑔)𝐺ℎ)) ↔ ∃𝑡(∃𝑔 ∈ (2nd ‘𝐵)∃ℎ ∈ (2nd ‘𝐶)𝑡 = (𝑔𝐺ℎ) ∧ 𝑥 = (𝑓𝐺𝑡))))
32		genpelvl.2	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝑦 ∈ Q ∧ 𝑧 ∈ Q) → (𝑦𝐺𝑧) ∈ Q)
33	32	caovcl 5925	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝑔 ∈ Q ∧ ℎ ∈ Q) → (𝑔𝐺ℎ) ∈ Q)
34		elisset 2700	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝑔𝐺ℎ) ∈ Q → ∃𝑡 𝑡 = (𝑔𝐺ℎ))
35	33, 34	syl 14	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝑔 ∈ Q ∧ ℎ ∈ Q) → ∃𝑡 𝑡 = (𝑔𝐺ℎ))
36	35	biantrurd 303	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑔 ∈ Q ∧ ℎ ∈ Q) → (𝑥 = ((𝑓𝐺𝑔)𝐺ℎ) ↔ (∃𝑡 𝑡 = (𝑔𝐺ℎ) ∧ 𝑥 = ((𝑓𝐺𝑔)𝐺ℎ))))
37		19.41v 1874	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (∃𝑡(𝑡 = (𝑔𝐺ℎ) ∧ 𝑥 = ((𝑓𝐺𝑔)𝐺ℎ)) ↔ (∃𝑡 𝑡 = (𝑔𝐺ℎ) ∧ 𝑥 = ((𝑓𝐺𝑔)𝐺ℎ)))
38	36, 37	syl6bbr 197	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑔 ∈ Q ∧ ℎ ∈ Q) → (𝑥 = ((𝑓𝐺𝑔)𝐺ℎ) ↔ ∃𝑡(𝑡 = (𝑔𝐺ℎ) ∧ 𝑥 = ((𝑓𝐺𝑔)𝐺ℎ))))
39	10, 38	sylan2 284	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑔 ∈ Q ∧ (𝐶 ∈ P ∧ ℎ ∈ (2nd ‘𝐶))) → (𝑥 = ((𝑓𝐺𝑔)𝐺ℎ) ↔ ∃𝑡(𝑡 = (𝑔𝐺ℎ) ∧ 𝑥 = ((𝑓𝐺𝑔)𝐺ℎ))))
40	39	anassrs 397	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝑔 ∈ Q ∧ 𝐶 ∈ P) ∧ ℎ ∈ (2nd ‘𝐶)) → (𝑥 = ((𝑓𝐺𝑔)𝐺ℎ) ↔ ∃𝑡(𝑡 = (𝑔𝐺ℎ) ∧ 𝑥 = ((𝑓𝐺𝑔)𝐺ℎ))))
41	40	rexbidva 2434	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑔 ∈ Q ∧ 𝐶 ∈ P) → (∃ℎ ∈ (2nd ‘𝐶)𝑥 = ((𝑓𝐺𝑔)𝐺ℎ) ↔ ∃ℎ ∈ (2nd ‘𝐶)∃𝑡(𝑡 = (𝑔𝐺ℎ) ∧ 𝑥 = ((𝑓𝐺𝑔)𝐺ℎ))))
42		rexcom4 2709	. . . . . . . . . . . . . . . . 17 ⊢ (∃ℎ ∈ (2nd ‘𝐶)∃𝑡(𝑡 = (𝑔𝐺ℎ) ∧ 𝑥 = ((𝑓𝐺𝑔)𝐺ℎ)) ↔ ∃𝑡∃ℎ ∈ (2nd ‘𝐶)(𝑡 = (𝑔𝐺ℎ) ∧ 𝑥 = ((𝑓𝐺𝑔)𝐺ℎ)))
43	41, 42	syl6bb 195	. . . . . . . . . . . . . . . 16 ⊢ ((𝑔 ∈ Q ∧ 𝐶 ∈ P) → (∃ℎ ∈ (2nd ‘𝐶)𝑥 = ((𝑓𝐺𝑔)𝐺ℎ) ↔ ∃𝑡∃ℎ ∈ (2nd ‘𝐶)(𝑡 = (𝑔𝐺ℎ) ∧ 𝑥 = ((𝑓𝐺𝑔)𝐺ℎ))))
44	43	ancoms 266	. . . . . . . . . . . . . . 15 ⊢ ((𝐶 ∈ P ∧ 𝑔 ∈ Q) → (∃ℎ ∈ (2nd ‘𝐶)𝑥 = ((𝑓𝐺𝑔)𝐺ℎ) ↔ ∃𝑡∃ℎ ∈ (2nd ‘𝐶)(𝑡 = (𝑔𝐺ℎ) ∧ 𝑥 = ((𝑓𝐺𝑔)𝐺ℎ))))
45	6, 44	sylan2 284	. . . . . . . . . . . . . 14 ⊢ ((𝐶 ∈ P ∧ (𝐵 ∈ P ∧ 𝑔 ∈ (2nd ‘𝐵))) → (∃ℎ ∈ (2nd ‘𝐶)𝑥 = ((𝑓𝐺𝑔)𝐺ℎ) ↔ ∃𝑡∃ℎ ∈ (2nd ‘𝐶)(𝑡 = (𝑔𝐺ℎ) ∧ 𝑥 = ((𝑓𝐺𝑔)𝐺ℎ))))
46	45	anassrs 397	. . . . . . . . . . . . 13 ⊢ (((𝐶 ∈ P ∧ 𝐵 ∈ P) ∧ 𝑔 ∈ (2nd ‘𝐵)) → (∃ℎ ∈ (2nd ‘𝐶)𝑥 = ((𝑓𝐺𝑔)𝐺ℎ) ↔ ∃𝑡∃ℎ ∈ (2nd ‘𝐶)(𝑡 = (𝑔𝐺ℎ) ∧ 𝑥 = ((𝑓𝐺𝑔)𝐺ℎ))))
47	46	rexbidva 2434	. . . . . . . . . . . 12 ⊢ ((𝐶 ∈ P ∧ 𝐵 ∈ P) → (∃𝑔 ∈ (2nd ‘𝐵)∃ℎ ∈ (2nd ‘𝐶)𝑥 = ((𝑓𝐺𝑔)𝐺ℎ) ↔ ∃𝑔 ∈ (2nd ‘𝐵)∃𝑡∃ℎ ∈ (2nd ‘𝐶)(𝑡 = (𝑔𝐺ℎ) ∧ 𝑥 = ((𝑓𝐺𝑔)𝐺ℎ))))
48	47	ancoms 266	. . . . . . . . . . 11 ⊢ ((𝐵 ∈ P ∧ 𝐶 ∈ P) → (∃𝑔 ∈ (2nd ‘𝐵)∃ℎ ∈ (2nd ‘𝐶)𝑥 = ((𝑓𝐺𝑔)𝐺ℎ) ↔ ∃𝑔 ∈ (2nd ‘𝐵)∃𝑡∃ℎ ∈ (2nd ‘𝐶)(𝑡 = (𝑔𝐺ℎ) ∧ 𝑥 = ((𝑓𝐺𝑔)𝐺ℎ))))
49		rexcom4 2709	. . . . . . . . . . 11 ⊢ (∃𝑔 ∈ (2nd ‘𝐵)∃𝑡∃ℎ ∈ (2nd ‘𝐶)(𝑡 = (𝑔𝐺ℎ) ∧ 𝑥 = ((𝑓𝐺𝑔)𝐺ℎ)) ↔ ∃𝑡∃𝑔 ∈ (2nd ‘𝐵)∃ℎ ∈ (2nd ‘𝐶)(𝑡 = (𝑔𝐺ℎ) ∧ 𝑥 = ((𝑓𝐺𝑔)𝐺ℎ)))
50	48, 49	syl6bb 195	. . . . . . . . . 10 ⊢ ((𝐵 ∈ P ∧ 𝐶 ∈ P) → (∃𝑔 ∈ (2nd ‘𝐵)∃ℎ ∈ (2nd ‘𝐶)𝑥 = ((𝑓𝐺𝑔)𝐺ℎ) ↔ ∃𝑡∃𝑔 ∈ (2nd ‘𝐵)∃ℎ ∈ (2nd ‘𝐶)(𝑡 = (𝑔𝐺ℎ) ∧ 𝑥 = ((𝑓𝐺𝑔)𝐺ℎ))))
51	50	adantr 274	. . . . . . . . 9 ⊢ (((𝐵 ∈ P ∧ 𝐶 ∈ P) ∧ 𝑓 ∈ Q) → (∃𝑔 ∈ (2nd ‘𝐵)∃ℎ ∈ (2nd ‘𝐶)𝑥 = ((𝑓𝐺𝑔)𝐺ℎ) ↔ ∃𝑡∃𝑔 ∈ (2nd ‘𝐵)∃ℎ ∈ (2nd ‘𝐶)(𝑡 = (𝑔𝐺ℎ) ∧ 𝑥 = ((𝑓𝐺𝑔)𝐺ℎ))))
52		df-rex 2422	. . . . . . . . . . 11 ⊢ (∃𝑡 ∈ (2nd ‘(𝐵𝐹𝐶))𝑥 = (𝑓𝐺𝑡) ↔ ∃𝑡(𝑡 ∈ (2nd ‘(𝐵𝐹𝐶)) ∧ 𝑥 = (𝑓𝐺𝑡)))
53		genpelvl.1	. . . . . . . . . . . . . 14 ⊢ 𝐹 = (𝑤 ∈ P, 𝑣 ∈ P ↦ ⟨{𝑥 ∈ Q ∣ ∃𝑦 ∈ Q ∃𝑧 ∈ Q (𝑦 ∈ (1st ‘𝑤) ∧ 𝑧 ∈ (1st ‘𝑣) ∧ 𝑥 = (𝑦𝐺𝑧))}, {𝑥 ∈ Q ∣ ∃𝑦 ∈ Q ∃𝑧 ∈ Q (𝑦 ∈ (2nd ‘𝑤) ∧ 𝑧 ∈ (2nd ‘𝑣) ∧ 𝑥 = (𝑦𝐺𝑧))}⟩)
54	53, 32	genpelvu 7321	. . . . . . . . . . . . 13 ⊢ ((𝐵 ∈ P ∧ 𝐶 ∈ P) → (𝑡 ∈ (2nd ‘(𝐵𝐹𝐶)) ↔ ∃𝑔 ∈ (2nd ‘𝐵)∃ℎ ∈ (2nd ‘𝐶)𝑡 = (𝑔𝐺ℎ)))
55	54	anbi1d 460	. . . . . . . . . . . 12 ⊢ ((𝐵 ∈ P ∧ 𝐶 ∈ P) → ((𝑡 ∈ (2nd ‘(𝐵𝐹𝐶)) ∧ 𝑥 = (𝑓𝐺𝑡)) ↔ (∃𝑔 ∈ (2nd ‘𝐵)∃ℎ ∈ (2nd ‘𝐶)𝑡 = (𝑔𝐺ℎ) ∧ 𝑥 = (𝑓𝐺𝑡))))
56	55	exbidv 1797	. . . . . . . . . . 11 ⊢ ((𝐵 ∈ P ∧ 𝐶 ∈ P) → (∃𝑡(𝑡 ∈ (2nd ‘(𝐵𝐹𝐶)) ∧ 𝑥 = (𝑓𝐺𝑡)) ↔ ∃𝑡(∃𝑔 ∈ (2nd ‘𝐵)∃ℎ ∈ (2nd ‘𝐶)𝑡 = (𝑔𝐺ℎ) ∧ 𝑥 = (𝑓𝐺𝑡))))
57	52, 56	syl5bb 191	. . . . . . . . . 10 ⊢ ((𝐵 ∈ P ∧ 𝐶 ∈ P) → (∃𝑡 ∈ (2nd ‘(𝐵𝐹𝐶))𝑥 = (𝑓𝐺𝑡) ↔ ∃𝑡(∃𝑔 ∈ (2nd ‘𝐵)∃ℎ ∈ (2nd ‘𝐶)𝑡 = (𝑔𝐺ℎ) ∧ 𝑥 = (𝑓𝐺𝑡))))
58	57	adantr 274	. . . . . . . . 9 ⊢ (((𝐵 ∈ P ∧ 𝐶 ∈ P) ∧ 𝑓 ∈ Q) → (∃𝑡 ∈ (2nd ‘(𝐵𝐹𝐶))𝑥 = (𝑓𝐺𝑡) ↔ ∃𝑡(∃𝑔 ∈ (2nd ‘𝐵)∃ℎ ∈ (2nd ‘𝐶)𝑡 = (𝑔𝐺ℎ) ∧ 𝑥 = (𝑓𝐺𝑡))))
59	31, 51, 58	3bitr4rd 220	. . . . . . . 8 ⊢ (((𝐵 ∈ P ∧ 𝐶 ∈ P) ∧ 𝑓 ∈ Q) → (∃𝑡 ∈ (2nd ‘(𝐵𝐹𝐶))𝑥 = (𝑓𝐺𝑡) ↔ ∃𝑔 ∈ (2nd ‘𝐵)∃ℎ ∈ (2nd ‘𝐶)𝑥 = ((𝑓𝐺𝑔)𝐺ℎ)))
60	3, 59	sylan2 284	. . . . . . 7 ⊢ (((𝐵 ∈ P ∧ 𝐶 ∈ P) ∧ (𝐴 ∈ P ∧ 𝑓 ∈ (2nd ‘𝐴))) → (∃𝑡 ∈ (2nd ‘(𝐵𝐹𝐶))𝑥 = (𝑓𝐺𝑡) ↔ ∃𝑔 ∈ (2nd ‘𝐵)∃ℎ ∈ (2nd ‘𝐶)𝑥 = ((𝑓𝐺𝑔)𝐺ℎ)))
61	60	anassrs 397	. . . . . 6 ⊢ ((((𝐵 ∈ P ∧ 𝐶 ∈ P) ∧ 𝐴 ∈ P) ∧ 𝑓 ∈ (2nd ‘𝐴)) → (∃𝑡 ∈ (2nd ‘(𝐵𝐹𝐶))𝑥 = (𝑓𝐺𝑡) ↔ ∃𝑔 ∈ (2nd ‘𝐵)∃ℎ ∈ (2nd ‘𝐶)𝑥 = ((𝑓𝐺𝑔)𝐺ℎ)))
62	61	rexbidva 2434	. . . . 5 ⊢ (((𝐵 ∈ P ∧ 𝐶 ∈ P) ∧ 𝐴 ∈ P) → (∃𝑓 ∈ (2nd ‘𝐴)∃𝑡 ∈ (2nd ‘(𝐵𝐹𝐶))𝑥 = (𝑓𝐺𝑡) ↔ ∃𝑓 ∈ (2nd ‘𝐴)∃𝑔 ∈ (2nd ‘𝐵)∃ℎ ∈ (2nd ‘𝐶)𝑥 = ((𝑓𝐺𝑔)𝐺ℎ)))
63	62	ancoms 266	. . . 4 ⊢ ((𝐴 ∈ P ∧ (𝐵 ∈ P ∧ 𝐶 ∈ P)) → (∃𝑓 ∈ (2nd ‘𝐴)∃𝑡 ∈ (2nd ‘(𝐵𝐹𝐶))𝑥 = (𝑓𝐺𝑡) ↔ ∃𝑓 ∈ (2nd ‘𝐴)∃𝑔 ∈ (2nd ‘𝐵)∃ℎ ∈ (2nd ‘𝐶)𝑥 = ((𝑓𝐺𝑔)𝐺ℎ)))
64	63	3impb 1177	. . 3 ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P ∧ 𝐶 ∈ P) → (∃𝑓 ∈ (2nd ‘𝐴)∃𝑡 ∈ (2nd ‘(𝐵𝐹𝐶))𝑥 = (𝑓𝐺𝑡) ↔ ∃𝑓 ∈ (2nd ‘𝐴)∃𝑔 ∈ (2nd ‘𝐵)∃ℎ ∈ (2nd ‘𝐶)𝑥 = ((𝑓𝐺𝑔)𝐺ℎ)))
65		genpassg.5	. . . . . 6 ⊢ ((𝑓 ∈ P ∧ 𝑔 ∈ P) → (𝑓𝐹𝑔) ∈ P)
66	65	caovcl 5925	. . . . 5 ⊢ ((𝐵 ∈ P ∧ 𝐶 ∈ P) → (𝐵𝐹𝐶) ∈ P)
67	53, 32	genpelvu 7321	. . . . 5 ⊢ ((𝐴 ∈ P ∧ (𝐵𝐹𝐶) ∈ P) → (𝑥 ∈ (2nd ‘(𝐴𝐹(𝐵𝐹𝐶))) ↔ ∃𝑓 ∈ (2nd ‘𝐴)∃𝑡 ∈ (2nd ‘(𝐵𝐹𝐶))𝑥 = (𝑓𝐺𝑡)))
68	66, 67	sylan2 284	. . . 4 ⊢ ((𝐴 ∈ P ∧ (𝐵 ∈ P ∧ 𝐶 ∈ P)) → (𝑥 ∈ (2nd ‘(𝐴𝐹(𝐵𝐹𝐶))) ↔ ∃𝑓 ∈ (2nd ‘𝐴)∃𝑡 ∈ (2nd ‘(𝐵𝐹𝐶))𝑥 = (𝑓𝐺𝑡)))
69	68	3impb 1177	. . 3 ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P ∧ 𝐶 ∈ P) → (𝑥 ∈ (2nd ‘(𝐴𝐹(𝐵𝐹𝐶))) ↔ ∃𝑓 ∈ (2nd ‘𝐴)∃𝑡 ∈ (2nd ‘(𝐵𝐹𝐶))𝑥 = (𝑓𝐺𝑡)))
70		df-rex 2422	. . . . 5 ⊢ (∃𝑡 ∈ (2nd ‘(𝐴𝐹𝐵))∃ℎ ∈ (2nd ‘𝐶)𝑥 = (𝑡𝐺ℎ) ↔ ∃𝑡(𝑡 ∈ (2nd ‘(𝐴𝐹𝐵)) ∧ ∃ℎ ∈ (2nd ‘𝐶)𝑥 = (𝑡𝐺ℎ)))
71	53, 32	genpelvu 7321	. . . . . . . 8 ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P) → (𝑡 ∈ (2nd ‘(𝐴𝐹𝐵)) ↔ ∃𝑓 ∈ (2nd ‘𝐴)∃𝑔 ∈ (2nd ‘𝐵)𝑡 = (𝑓𝐺𝑔)))
72	71	3adant3 1001	. . . . . . 7 ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P ∧ 𝐶 ∈ P) → (𝑡 ∈ (2nd ‘(𝐴𝐹𝐵)) ↔ ∃𝑓 ∈ (2nd ‘𝐴)∃𝑔 ∈ (2nd ‘𝐵)𝑡 = (𝑓𝐺𝑔)))
73	72	anbi1d 460	. . . . . 6 ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P ∧ 𝐶 ∈ P) → ((𝑡 ∈ (2nd ‘(𝐴𝐹𝐵)) ∧ ∃ℎ ∈ (2nd ‘𝐶)𝑥 = (𝑡𝐺ℎ)) ↔ (∃𝑓 ∈ (2nd ‘𝐴)∃𝑔 ∈ (2nd ‘𝐵)𝑡 = (𝑓𝐺𝑔) ∧ ∃ℎ ∈ (2nd ‘𝐶)𝑥 = (𝑡𝐺ℎ))))
74	73	exbidv 1797	. . . . 5 ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P ∧ 𝐶 ∈ P) → (∃𝑡(𝑡 ∈ (2nd ‘(𝐴𝐹𝐵)) ∧ ∃ℎ ∈ (2nd ‘𝐶)𝑥 = (𝑡𝐺ℎ)) ↔ ∃𝑡(∃𝑓 ∈ (2nd ‘𝐴)∃𝑔 ∈ (2nd ‘𝐵)𝑡 = (𝑓𝐺𝑔) ∧ ∃ℎ ∈ (2nd ‘𝐶)𝑥 = (𝑡𝐺ℎ))))
75	70, 74	syl5bb 191	. . . 4 ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P ∧ 𝐶 ∈ P) → (∃𝑡 ∈ (2nd ‘(𝐴𝐹𝐵))∃ℎ ∈ (2nd ‘𝐶)𝑥 = (𝑡𝐺ℎ) ↔ ∃𝑡(∃𝑓 ∈ (2nd ‘𝐴)∃𝑔 ∈ (2nd ‘𝐵)𝑡 = (𝑓𝐺𝑔) ∧ ∃ℎ ∈ (2nd ‘𝐶)𝑥 = (𝑡𝐺ℎ))))
76	65	caovcl 5925	. . . . . 6 ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P) → (𝐴𝐹𝐵) ∈ P)
77	53, 32	genpelvu 7321	. . . . . 6 ⊢ (((𝐴𝐹𝐵) ∈ P ∧ 𝐶 ∈ P) → (𝑥 ∈ (2nd ‘((𝐴𝐹𝐵)𝐹𝐶)) ↔ ∃𝑡 ∈ (2nd ‘(𝐴𝐹𝐵))∃ℎ ∈ (2nd ‘𝐶)𝑥 = (𝑡𝐺ℎ)))
78	76, 77	sylan 281	. . . . 5 ⊢ (((𝐴 ∈ P ∧ 𝐵 ∈ P) ∧ 𝐶 ∈ P) → (𝑥 ∈ (2nd ‘((𝐴𝐹𝐵)𝐹𝐶)) ↔ ∃𝑡 ∈ (2nd ‘(𝐴𝐹𝐵))∃ℎ ∈ (2nd ‘𝐶)𝑥 = (𝑡𝐺ℎ)))
79	78	3impa 1176	. . . 4 ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P ∧ 𝐶 ∈ P) → (𝑥 ∈ (2nd ‘((𝐴𝐹𝐵)𝐹𝐶)) ↔ ∃𝑡 ∈ (2nd ‘(𝐴𝐹𝐵))∃ℎ ∈ (2nd ‘𝐶)𝑥 = (𝑡𝐺ℎ)))
80	32	caovcl 5925	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑓 ∈ Q ∧ 𝑔 ∈ Q) → (𝑓𝐺𝑔) ∈ Q)
81		elisset 2700	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑓𝐺𝑔) ∈ Q → ∃𝑡 𝑡 = (𝑓𝐺𝑔))
82	80, 81	syl 14	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑓 ∈ Q ∧ 𝑔 ∈ Q) → ∃𝑡 𝑡 = (𝑓𝐺𝑔))
83	82	biantrurd 303	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑓 ∈ Q ∧ 𝑔 ∈ Q) → (∃ℎ ∈ (2nd ‘𝐶)𝑥 = ((𝑓𝐺𝑔)𝐺ℎ) ↔ (∃𝑡 𝑡 = (𝑓𝐺𝑔) ∧ ∃ℎ ∈ (2nd ‘𝐶)𝑥 = ((𝑓𝐺𝑔)𝐺ℎ))))
84		oveq1 5781	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑡 = (𝑓𝐺𝑔) → (𝑡𝐺ℎ) = ((𝑓𝐺𝑔)𝐺ℎ))
85	84	eqeq2d 2151	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑡 = (𝑓𝐺𝑔) → (𝑥 = (𝑡𝐺ℎ) ↔ 𝑥 = ((𝑓𝐺𝑔)𝐺ℎ)))
86	85	rexbidv 2438	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑡 = (𝑓𝐺𝑔) → (∃ℎ ∈ (2nd ‘𝐶)𝑥 = (𝑡𝐺ℎ) ↔ ∃ℎ ∈ (2nd ‘𝐶)𝑥 = ((𝑓𝐺𝑔)𝐺ℎ)))
87	86	pm5.32i 449	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑡 = (𝑓𝐺𝑔) ∧ ∃ℎ ∈ (2nd ‘𝐶)𝑥 = (𝑡𝐺ℎ)) ↔ (𝑡 = (𝑓𝐺𝑔) ∧ ∃ℎ ∈ (2nd ‘𝐶)𝑥 = ((𝑓𝐺𝑔)𝐺ℎ)))
88	87	exbii 1584	. . . . . . . . . . . . . . . . . 18 ⊢ (∃𝑡(𝑡 = (𝑓𝐺𝑔) ∧ ∃ℎ ∈ (2nd ‘𝐶)𝑥 = (𝑡𝐺ℎ)) ↔ ∃𝑡(𝑡 = (𝑓𝐺𝑔) ∧ ∃ℎ ∈ (2nd ‘𝐶)𝑥 = ((𝑓𝐺𝑔)𝐺ℎ)))
89		19.41v 1874	. . . . . . . . . . . . . . . . . 18 ⊢ (∃𝑡(𝑡 = (𝑓𝐺𝑔) ∧ ∃ℎ ∈ (2nd ‘𝐶)𝑥 = ((𝑓𝐺𝑔)𝐺ℎ)) ↔ (∃𝑡 𝑡 = (𝑓𝐺𝑔) ∧ ∃ℎ ∈ (2nd ‘𝐶)𝑥 = ((𝑓𝐺𝑔)𝐺ℎ)))
90	88, 89	bitri 183	. . . . . . . . . . . . . . . . 17 ⊢ (∃𝑡(𝑡 = (𝑓𝐺𝑔) ∧ ∃ℎ ∈ (2nd ‘𝐶)𝑥 = (𝑡𝐺ℎ)) ↔ (∃𝑡 𝑡 = (𝑓𝐺𝑔) ∧ ∃ℎ ∈ (2nd ‘𝐶)𝑥 = ((𝑓𝐺𝑔)𝐺ℎ)))
91	83, 90	syl6bbr 197	. . . . . . . . . . . . . . . 16 ⊢ ((𝑓 ∈ Q ∧ 𝑔 ∈ Q) → (∃ℎ ∈ (2nd ‘𝐶)𝑥 = ((𝑓𝐺𝑔)𝐺ℎ) ↔ ∃𝑡(𝑡 = (𝑓𝐺𝑔) ∧ ∃ℎ ∈ (2nd ‘𝐶)𝑥 = (𝑡𝐺ℎ))))
92	6, 91	sylan2 284	. . . . . . . . . . . . . . 15 ⊢ ((𝑓 ∈ Q ∧ (𝐵 ∈ P ∧ 𝑔 ∈ (2nd ‘𝐵))) → (∃ℎ ∈ (2nd ‘𝐶)𝑥 = ((𝑓𝐺𝑔)𝐺ℎ) ↔ ∃𝑡(𝑡 = (𝑓𝐺𝑔) ∧ ∃ℎ ∈ (2nd ‘𝐶)𝑥 = (𝑡𝐺ℎ))))
93	92	anassrs 397	. . . . . . . . . . . . . 14 ⊢ (((𝑓 ∈ Q ∧ 𝐵 ∈ P) ∧ 𝑔 ∈ (2nd ‘𝐵)) → (∃ℎ ∈ (2nd ‘𝐶)𝑥 = ((𝑓𝐺𝑔)𝐺ℎ) ↔ ∃𝑡(𝑡 = (𝑓𝐺𝑔) ∧ ∃ℎ ∈ (2nd ‘𝐶)𝑥 = (𝑡𝐺ℎ))))
94	93	rexbidva 2434	. . . . . . . . . . . . 13 ⊢ ((𝑓 ∈ Q ∧ 𝐵 ∈ P) → (∃𝑔 ∈ (2nd ‘𝐵)∃ℎ ∈ (2nd ‘𝐶)𝑥 = ((𝑓𝐺𝑔)𝐺ℎ) ↔ ∃𝑔 ∈ (2nd ‘𝐵)∃𝑡(𝑡 = (𝑓𝐺𝑔) ∧ ∃ℎ ∈ (2nd ‘𝐶)𝑥 = (𝑡𝐺ℎ))))
95		rexcom4 2709	. . . . . . . . . . . . 13 ⊢ (∃𝑔 ∈ (2nd ‘𝐵)∃𝑡(𝑡 = (𝑓𝐺𝑔) ∧ ∃ℎ ∈ (2nd ‘𝐶)𝑥 = (𝑡𝐺ℎ)) ↔ ∃𝑡∃𝑔 ∈ (2nd ‘𝐵)(𝑡 = (𝑓𝐺𝑔) ∧ ∃ℎ ∈ (2nd ‘𝐶)𝑥 = (𝑡𝐺ℎ)))
96	94, 95	syl6bb 195	. . . . . . . . . . . 12 ⊢ ((𝑓 ∈ Q ∧ 𝐵 ∈ P) → (∃𝑔 ∈ (2nd ‘𝐵)∃ℎ ∈ (2nd ‘𝐶)𝑥 = ((𝑓𝐺𝑔)𝐺ℎ) ↔ ∃𝑡∃𝑔 ∈ (2nd ‘𝐵)(𝑡 = (𝑓𝐺𝑔) ∧ ∃ℎ ∈ (2nd ‘𝐶)𝑥 = (𝑡𝐺ℎ))))
97	96	ancoms 266	. . . . . . . . . . 11 ⊢ ((𝐵 ∈ P ∧ 𝑓 ∈ Q) → (∃𝑔 ∈ (2nd ‘𝐵)∃ℎ ∈ (2nd ‘𝐶)𝑥 = ((𝑓𝐺𝑔)𝐺ℎ) ↔ ∃𝑡∃𝑔 ∈ (2nd ‘𝐵)(𝑡 = (𝑓𝐺𝑔) ∧ ∃ℎ ∈ (2nd ‘𝐶)𝑥 = (𝑡𝐺ℎ))))
98	3, 97	sylan2 284	. . . . . . . . . 10 ⊢ ((𝐵 ∈ P ∧ (𝐴 ∈ P ∧ 𝑓 ∈ (2nd ‘𝐴))) → (∃𝑔 ∈ (2nd ‘𝐵)∃ℎ ∈ (2nd ‘𝐶)𝑥 = ((𝑓𝐺𝑔)𝐺ℎ) ↔ ∃𝑡∃𝑔 ∈ (2nd ‘𝐵)(𝑡 = (𝑓𝐺𝑔) ∧ ∃ℎ ∈ (2nd ‘𝐶)𝑥 = (𝑡𝐺ℎ))))
99	98	anassrs 397	. . . . . . . . 9 ⊢ (((𝐵 ∈ P ∧ 𝐴 ∈ P) ∧ 𝑓 ∈ (2nd ‘𝐴)) → (∃𝑔 ∈ (2nd ‘𝐵)∃ℎ ∈ (2nd ‘𝐶)𝑥 = ((𝑓𝐺𝑔)𝐺ℎ) ↔ ∃𝑡∃𝑔 ∈ (2nd ‘𝐵)(𝑡 = (𝑓𝐺𝑔) ∧ ∃ℎ ∈ (2nd ‘𝐶)𝑥 = (𝑡𝐺ℎ))))
100	99	rexbidva 2434	. . . . . . . 8 ⊢ ((𝐵 ∈ P ∧ 𝐴 ∈ P) → (∃𝑓 ∈ (2nd ‘𝐴)∃𝑔 ∈ (2nd ‘𝐵)∃ℎ ∈ (2nd ‘𝐶)𝑥 = ((𝑓𝐺𝑔)𝐺ℎ) ↔ ∃𝑓 ∈ (2nd ‘𝐴)∃𝑡∃𝑔 ∈ (2nd ‘𝐵)(𝑡 = (𝑓𝐺𝑔) ∧ ∃ℎ ∈ (2nd ‘𝐶)𝑥 = (𝑡𝐺ℎ))))
101		rexcom4 2709	. . . . . . . 8 ⊢ (∃𝑓 ∈ (2nd ‘𝐴)∃𝑡∃𝑔 ∈ (2nd ‘𝐵)(𝑡 = (𝑓𝐺𝑔) ∧ ∃ℎ ∈ (2nd ‘𝐶)𝑥 = (𝑡𝐺ℎ)) ↔ ∃𝑡∃𝑓 ∈ (2nd ‘𝐴)∃𝑔 ∈ (2nd ‘𝐵)(𝑡 = (𝑓𝐺𝑔) ∧ ∃ℎ ∈ (2nd ‘𝐶)𝑥 = (𝑡𝐺ℎ)))
102	100, 101	syl6bb 195	. . . . . . 7 ⊢ ((𝐵 ∈ P ∧ 𝐴 ∈ P) → (∃𝑓 ∈ (2nd ‘𝐴)∃𝑔 ∈ (2nd ‘𝐵)∃ℎ ∈ (2nd ‘𝐶)𝑥 = ((𝑓𝐺𝑔)𝐺ℎ) ↔ ∃𝑡∃𝑓 ∈ (2nd ‘𝐴)∃𝑔 ∈ (2nd ‘𝐵)(𝑡 = (𝑓𝐺𝑔) ∧ ∃ℎ ∈ (2nd ‘𝐶)𝑥 = (𝑡𝐺ℎ))))
103		r19.41v 2587	. . . . . . . . . 10 ⊢ (∃𝑔 ∈ (2nd ‘𝐵)(𝑡 = (𝑓𝐺𝑔) ∧ ∃ℎ ∈ (2nd ‘𝐶)𝑥 = (𝑡𝐺ℎ)) ↔ (∃𝑔 ∈ (2nd ‘𝐵)𝑡 = (𝑓𝐺𝑔) ∧ ∃ℎ ∈ (2nd ‘𝐶)𝑥 = (𝑡𝐺ℎ)))
104	103	rexbii 2442	. . . . . . . . 9 ⊢ (∃𝑓 ∈ (2nd ‘𝐴)∃𝑔 ∈ (2nd ‘𝐵)(𝑡 = (𝑓𝐺𝑔) ∧ ∃ℎ ∈ (2nd ‘𝐶)𝑥 = (𝑡𝐺ℎ)) ↔ ∃𝑓 ∈ (2nd ‘𝐴)(∃𝑔 ∈ (2nd ‘𝐵)𝑡 = (𝑓𝐺𝑔) ∧ ∃ℎ ∈ (2nd ‘𝐶)𝑥 = (𝑡𝐺ℎ)))
105		r19.41v 2587	. . . . . . . . 9 ⊢ (∃𝑓 ∈ (2nd ‘𝐴)(∃𝑔 ∈ (2nd ‘𝐵)𝑡 = (𝑓𝐺𝑔) ∧ ∃ℎ ∈ (2nd ‘𝐶)𝑥 = (𝑡𝐺ℎ)) ↔ (∃𝑓 ∈ (2nd ‘𝐴)∃𝑔 ∈ (2nd ‘𝐵)𝑡 = (𝑓𝐺𝑔) ∧ ∃ℎ ∈ (2nd ‘𝐶)𝑥 = (𝑡𝐺ℎ)))
106	104, 105	bitri 183	. . . . . . . 8 ⊢ (∃𝑓 ∈ (2nd ‘𝐴)∃𝑔 ∈ (2nd ‘𝐵)(𝑡 = (𝑓𝐺𝑔) ∧ ∃ℎ ∈ (2nd ‘𝐶)𝑥 = (𝑡𝐺ℎ)) ↔ (∃𝑓 ∈ (2nd ‘𝐴)∃𝑔 ∈ (2nd ‘𝐵)𝑡 = (𝑓𝐺𝑔) ∧ ∃ℎ ∈ (2nd ‘𝐶)𝑥 = (𝑡𝐺ℎ)))
107	106	exbii 1584	. . . . . . 7 ⊢ (∃𝑡∃𝑓 ∈ (2nd ‘𝐴)∃𝑔 ∈ (2nd ‘𝐵)(𝑡 = (𝑓𝐺𝑔) ∧ ∃ℎ ∈ (2nd ‘𝐶)𝑥 = (𝑡𝐺ℎ)) ↔ ∃𝑡(∃𝑓 ∈ (2nd ‘𝐴)∃𝑔 ∈ (2nd ‘𝐵)𝑡 = (𝑓𝐺𝑔) ∧ ∃ℎ ∈ (2nd ‘𝐶)𝑥 = (𝑡𝐺ℎ)))
108	102, 107	syl6bb 195	. . . . . 6 ⊢ ((𝐵 ∈ P ∧ 𝐴 ∈ P) → (∃𝑓 ∈ (2nd ‘𝐴)∃𝑔 ∈ (2nd ‘𝐵)∃ℎ ∈ (2nd ‘𝐶)𝑥 = ((𝑓𝐺𝑔)𝐺ℎ) ↔ ∃𝑡(∃𝑓 ∈ (2nd ‘𝐴)∃𝑔 ∈ (2nd ‘𝐵)𝑡 = (𝑓𝐺𝑔) ∧ ∃ℎ ∈ (2nd ‘𝐶)𝑥 = (𝑡𝐺ℎ))))
109	108	ancoms 266	. . . . 5 ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P) → (∃𝑓 ∈ (2nd ‘𝐴)∃𝑔 ∈ (2nd ‘𝐵)∃ℎ ∈ (2nd ‘𝐶)𝑥 = ((𝑓𝐺𝑔)𝐺ℎ) ↔ ∃𝑡(∃𝑓 ∈ (2nd ‘𝐴)∃𝑔 ∈ (2nd ‘𝐵)𝑡 = (𝑓𝐺𝑔) ∧ ∃ℎ ∈ (2nd ‘𝐶)𝑥 = (𝑡𝐺ℎ))))
110	109	3adant3 1001	. . . 4 ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P ∧ 𝐶 ∈ P) → (∃𝑓 ∈ (2nd ‘𝐴)∃𝑔 ∈ (2nd ‘𝐵)∃ℎ ∈ (2nd ‘𝐶)𝑥 = ((𝑓𝐺𝑔)𝐺ℎ) ↔ ∃𝑡(∃𝑓 ∈ (2nd ‘𝐴)∃𝑔 ∈ (2nd ‘𝐵)𝑡 = (𝑓𝐺𝑔) ∧ ∃ℎ ∈ (2nd ‘𝐶)𝑥 = (𝑡𝐺ℎ))))
111	75, 79, 110	3bitr4d 219	. . 3 ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P ∧ 𝐶 ∈ P) → (𝑥 ∈ (2nd ‘((𝐴𝐹𝐵)𝐹𝐶)) ↔ ∃𝑓 ∈ (2nd ‘𝐴)∃𝑔 ∈ (2nd ‘𝐵)∃ℎ ∈ (2nd ‘𝐶)𝑥 = ((𝑓𝐺𝑔)𝐺ℎ)))
112	64, 69, 111	3bitr4rd 220	. 2 ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P ∧ 𝐶 ∈ P) → (𝑥 ∈ (2nd ‘((𝐴𝐹𝐵)𝐹𝐶)) ↔ 𝑥 ∈ (2nd ‘(𝐴𝐹(𝐵𝐹𝐶)))))
113	112	eqrdv 2137	1 ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P ∧ 𝐶 ∈ P) → (2nd ‘((𝐴𝐹𝐵)𝐹𝐶)) = (2nd ‘(𝐴𝐹(𝐵𝐹𝐶))))

Syntax hints:   → wi 4   ∧ wa 103   ↔ wb 104   ∧ w3a 962   = wceq 1331  ∃wex 1468   ∈ wcel 1480  ∃wrex 2417  {crab 2420  ⟨cop 3530   × cxp 4537  dom cdm 4539  ‘cfv 5123  (class class class)co 5774   ∈ cmpo 5776  1^st c1st 6036  2^nd c2nd 6037  Qcnq 7088  Pcnp 7099

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-13 1491  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-coll 4043  ax-sep 4046  ax-pow 4098  ax-pr 4131  ax-un 4355  ax-setind 4452  ax-iinf 4502

This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-fal 1337  df-nf 1437  df-sb 1736  df-eu 2002  df-mo 2003  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ne 2309  df-ral 2421  df-rex 2422  df-reu 2423  df-rab 2425  df-v 2688  df-sbc 2910  df-csb 3004  df-dif 3073  df-un 3075  df-in 3077  df-ss 3084  df-pw 3512  df-sn 3533  df-pr 3534  df-op 3536  df-uni 3737  df-int 3772  df-iun 3815  df-br 3930  df-opab 3990  df-mpt 3991  df-id 4215  df-iom 4505  df-xp 4545  df-rel 4546  df-cnv 4547  df-co 4548  df-dm 4549  df-rn 4550  df-res 4551  df-ima 4552  df-iota 5088  df-fun 5125  df-fn 5126  df-f 5127  df-f1 5128  df-fo 5129  df-f1o 5130  df-fv 5131  df-ov 5777  df-oprab 5778  df-mpo 5779  df-1st 6038  df-2nd 6039  df-qs 6435  df-ni 7112  df-nqqs 7156  df-inp 7274

This theorem is referenced by:  genpassg  7334