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Theorem genpassl 7456

Description: Associativity of lower cuts. Lemma for genpassg 7458. (Contributed by Jim Kingdon, 11-Dec-2019.)

**Hypotheses**
Ref	Expression
genpelvl.1	⊢ 𝐹 = (𝑤 ∈ P, 𝑣 ∈ P ↦ ⟨{𝑥 ∈ Q ∣ ∃𝑦 ∈ Q ∃𝑧 ∈ Q (𝑦 ∈ (1^st ‘𝑤) ∧ 𝑧 ∈ (1^st ‘𝑣) ∧ 𝑥 = (𝑦𝐺𝑧))}, {𝑥 ∈ Q ∣ ∃𝑦 ∈ Q ∃𝑧 ∈ Q (𝑦 ∈ (2^nd ‘𝑤) ∧ 𝑧 ∈ (2^nd ‘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
genpassl	⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P ∧ 𝐶 ∈ P) → (1^st ‘((𝐴𝐹𝐵)𝐹𝐶)) = (1^st ‘(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 genpassl Dummy variable 𝑡 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		prop 7407	. . . . . . . . 9 ⊢ (𝐴 ∈ P → ⟨(1^st ‘𝐴), (2^nd ‘𝐴)⟩ ∈ P)
2		elprnql 7413	. . . . . . . . 9 ⊢ ((⟨(1^st ‘𝐴), (2^nd ‘𝐴)⟩ ∈ P ∧ 𝑓 ∈ (1^st ‘𝐴)) → 𝑓 ∈ Q)
3	1, 2	sylan 281	. . . . . . . 8 ⊢ ((𝐴 ∈ P ∧ 𝑓 ∈ (1^st ‘𝐴)) → 𝑓 ∈ Q)
4		prop 7407	. . . . . . . . . . . . . . . 16 ⊢ (𝐵 ∈ P → ⟨(1^st ‘𝐵), (2^nd ‘𝐵)⟩ ∈ P)
5		elprnql 7413	. . . . . . . . . . . . . . . 16 ⊢ ((⟨(1^st ‘𝐵), (2^nd ‘𝐵)⟩ ∈ P ∧ 𝑔 ∈ (1^st ‘𝐵)) → 𝑔 ∈ Q)
6	4, 5	sylan 281	. . . . . . . . . . . . . . 15 ⊢ ((𝐵 ∈ P ∧ 𝑔 ∈ (1^st ‘𝐵)) → 𝑔 ∈ Q)
7		prop 7407	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝐶 ∈ P → ⟨(1^st ‘𝐶), (2^nd ‘𝐶)⟩ ∈ P)
8		elprnql 7413	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((⟨(1^st ‘𝐶), (2^nd ‘𝐶)⟩ ∈ P ∧ ℎ ∈ (1^st ‘𝐶)) → ℎ ∈ Q)
9	7, 8	sylan 281	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝐶 ∈ P ∧ ℎ ∈ (1^st ‘𝐶)) → ℎ ∈ Q)
10		oveq2 5844	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝑡 = (𝑔𝐺ℎ) → (𝑓𝐺𝑡) = (𝑓𝐺(𝑔𝐺ℎ)))
11	10	adantr 274	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝑡 = (𝑔𝐺ℎ) ∧ (𝑓 ∈ Q ∧ 𝑔 ∈ Q ∧ ℎ ∈ Q)) → (𝑓𝐺𝑡) = (𝑓𝐺(𝑔𝐺ℎ)))
12		genpassg.6	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((𝑓 ∈ Q ∧ 𝑔 ∈ Q ∧ ℎ ∈ Q) → ((𝑓𝐺𝑔)𝐺ℎ) = (𝑓𝐺(𝑔𝐺ℎ)))
13	12	adantl 275	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝑡 = (𝑔𝐺ℎ) ∧ (𝑓 ∈ Q ∧ 𝑔 ∈ Q ∧ ℎ ∈ Q)) → ((𝑓𝐺𝑔)𝐺ℎ) = (𝑓𝐺(𝑔𝐺ℎ)))
14	11, 13	eqtr4d 2200	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝑡 = (𝑔𝐺ℎ) ∧ (𝑓 ∈ Q ∧ 𝑔 ∈ Q ∧ ℎ ∈ Q)) → (𝑓𝐺𝑡) = ((𝑓𝐺𝑔)𝐺ℎ))
15	14	eqeq2d 2176	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝑡 = (𝑔𝐺ℎ) ∧ (𝑓 ∈ Q ∧ 𝑔 ∈ Q ∧ ℎ ∈ Q)) → (𝑥 = (𝑓𝐺𝑡) ↔ 𝑥 = ((𝑓𝐺𝑔)𝐺ℎ)))
16	15	expcom 115	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝑓 ∈ Q ∧ 𝑔 ∈ Q ∧ ℎ ∈ Q) → (𝑡 = (𝑔𝐺ℎ) → (𝑥 = (𝑓𝐺𝑡) ↔ 𝑥 = ((𝑓𝐺𝑔)𝐺ℎ))))
17	16	pm5.32d 446	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑓 ∈ Q ∧ 𝑔 ∈ Q ∧ ℎ ∈ Q) → ((𝑡 = (𝑔𝐺ℎ) ∧ 𝑥 = (𝑓𝐺𝑡)) ↔ (𝑡 = (𝑔𝐺ℎ) ∧ 𝑥 = ((𝑓𝐺𝑔)𝐺ℎ))))
18	17	3expa 1192	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝑓 ∈ Q ∧ 𝑔 ∈ Q) ∧ ℎ ∈ Q) → ((𝑡 = (𝑔𝐺ℎ) ∧ 𝑥 = (𝑓𝐺𝑡)) ↔ (𝑡 = (𝑔𝐺ℎ) ∧ 𝑥 = ((𝑓𝐺𝑔)𝐺ℎ))))
19	9, 18	sylan2 284	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝑓 ∈ Q ∧ 𝑔 ∈ Q) ∧ (𝐶 ∈ P ∧ ℎ ∈ (1^st ‘𝐶))) → ((𝑡 = (𝑔𝐺ℎ) ∧ 𝑥 = (𝑓𝐺𝑡)) ↔ (𝑡 = (𝑔𝐺ℎ) ∧ 𝑥 = ((𝑓𝐺𝑔)𝐺ℎ))))
20	19	anassrs 398	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝑓 ∈ Q ∧ 𝑔 ∈ Q) ∧ 𝐶 ∈ P) ∧ ℎ ∈ (1^st ‘𝐶)) → ((𝑡 = (𝑔𝐺ℎ) ∧ 𝑥 = (𝑓𝐺𝑡)) ↔ (𝑡 = (𝑔𝐺ℎ) ∧ 𝑥 = ((𝑓𝐺𝑔)𝐺ℎ))))
21	20	rexbidva 2461	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑓 ∈ Q ∧ 𝑔 ∈ Q) ∧ 𝐶 ∈ P) → (∃ℎ ∈ (1^st ‘𝐶)(𝑡 = (𝑔𝐺ℎ) ∧ 𝑥 = (𝑓𝐺𝑡)) ↔ ∃ℎ ∈ (1^st ‘𝐶)(𝑡 = (𝑔𝐺ℎ) ∧ 𝑥 = ((𝑓𝐺𝑔)𝐺ℎ))))
22		r19.41v 2620	. . . . . . . . . . . . . . . . 17 ⊢ (∃ℎ ∈ (1^st ‘𝐶)(𝑡 = (𝑔𝐺ℎ) ∧ 𝑥 = (𝑓𝐺𝑡)) ↔ (∃ℎ ∈ (1^st ‘𝐶)𝑡 = (𝑔𝐺ℎ) ∧ 𝑥 = (𝑓𝐺𝑡)))
23	21, 22	bitr3di 194	. . . . . . . . . . . . . . . 16 ⊢ (((𝑓 ∈ Q ∧ 𝑔 ∈ Q) ∧ 𝐶 ∈ P) → (∃ℎ ∈ (1^st ‘𝐶)(𝑡 = (𝑔𝐺ℎ) ∧ 𝑥 = ((𝑓𝐺𝑔)𝐺ℎ)) ↔ (∃ℎ ∈ (1^st ‘𝐶)𝑡 = (𝑔𝐺ℎ) ∧ 𝑥 = (𝑓𝐺𝑡))))
24	23	an32s 558	. . . . . . . . . . . . . . 15 ⊢ (((𝑓 ∈ Q ∧ 𝐶 ∈ P) ∧ 𝑔 ∈ Q) → (∃ℎ ∈ (1^st ‘𝐶)(𝑡 = (𝑔𝐺ℎ) ∧ 𝑥 = ((𝑓𝐺𝑔)𝐺ℎ)) ↔ (∃ℎ ∈ (1^st ‘𝐶)𝑡 = (𝑔𝐺ℎ) ∧ 𝑥 = (𝑓𝐺𝑡))))
25	6, 24	sylan2 284	. . . . . . . . . . . . . 14 ⊢ (((𝑓 ∈ Q ∧ 𝐶 ∈ P) ∧ (𝐵 ∈ P ∧ 𝑔 ∈ (1^st ‘𝐵))) → (∃ℎ ∈ (1^st ‘𝐶)(𝑡 = (𝑔𝐺ℎ) ∧ 𝑥 = ((𝑓𝐺𝑔)𝐺ℎ)) ↔ (∃ℎ ∈ (1^st ‘𝐶)𝑡 = (𝑔𝐺ℎ) ∧ 𝑥 = (𝑓𝐺𝑡))))
26	25	anassrs 398	. . . . . . . . . . . . 13 ⊢ ((((𝑓 ∈ Q ∧ 𝐶 ∈ P) ∧ 𝐵 ∈ P) ∧ 𝑔 ∈ (1^st ‘𝐵)) → (∃ℎ ∈ (1^st ‘𝐶)(𝑡 = (𝑔𝐺ℎ) ∧ 𝑥 = ((𝑓𝐺𝑔)𝐺ℎ)) ↔ (∃ℎ ∈ (1^st ‘𝐶)𝑡 = (𝑔𝐺ℎ) ∧ 𝑥 = (𝑓𝐺𝑡))))
27	26	rexbidva 2461	. . . . . . . . . . . 12 ⊢ (((𝑓 ∈ Q ∧ 𝐶 ∈ P) ∧ 𝐵 ∈ P) → (∃𝑔 ∈ (1^st ‘𝐵)∃ℎ ∈ (1^st ‘𝐶)(𝑡 = (𝑔𝐺ℎ) ∧ 𝑥 = ((𝑓𝐺𝑔)𝐺ℎ)) ↔ ∃𝑔 ∈ (1^st ‘𝐵)(∃ℎ ∈ (1^st ‘𝐶)𝑡 = (𝑔𝐺ℎ) ∧ 𝑥 = (𝑓𝐺𝑡))))
28		r19.41v 2620	. . . . . . . . . . . 12 ⊢ (∃𝑔 ∈ (1^st ‘𝐵)(∃ℎ ∈ (1^st ‘𝐶)𝑡 = (𝑔𝐺ℎ) ∧ 𝑥 = (𝑓𝐺𝑡)) ↔ (∃𝑔 ∈ (1^st ‘𝐵)∃ℎ ∈ (1^st ‘𝐶)𝑡 = (𝑔𝐺ℎ) ∧ 𝑥 = (𝑓𝐺𝑡)))
29	27, 28	bitrdi 195	. . . . . . . . . . 11 ⊢ (((𝑓 ∈ Q ∧ 𝐶 ∈ P) ∧ 𝐵 ∈ P) → (∃𝑔 ∈ (1^st ‘𝐵)∃ℎ ∈ (1^st ‘𝐶)(𝑡 = (𝑔𝐺ℎ) ∧ 𝑥 = ((𝑓𝐺𝑔)𝐺ℎ)) ↔ (∃𝑔 ∈ (1^st ‘𝐵)∃ℎ ∈ (1^st ‘𝐶)𝑡 = (𝑔𝐺ℎ) ∧ 𝑥 = (𝑓𝐺𝑡))))
30	29	an31s 560	. . . . . . . . . 10 ⊢ (((𝐵 ∈ P ∧ 𝐶 ∈ P) ∧ 𝑓 ∈ Q) → (∃𝑔 ∈ (1^st ‘𝐵)∃ℎ ∈ (1^st ‘𝐶)(𝑡 = (𝑔𝐺ℎ) ∧ 𝑥 = ((𝑓𝐺𝑔)𝐺ℎ)) ↔ (∃𝑔 ∈ (1^st ‘𝐵)∃ℎ ∈ (1^st ‘𝐶)𝑡 = (𝑔𝐺ℎ) ∧ 𝑥 = (𝑓𝐺𝑡))))
31	30	exbidv 1812	. . . . . . . . 9 ⊢ (((𝐵 ∈ P ∧ 𝐶 ∈ P) ∧ 𝑓 ∈ Q) → (∃𝑡∃𝑔 ∈ (1^st ‘𝐵)∃ℎ ∈ (1^st ‘𝐶)(𝑡 = (𝑔𝐺ℎ) ∧ 𝑥 = ((𝑓𝐺𝑔)𝐺ℎ)) ↔ ∃𝑡(∃𝑔 ∈ (1^st ‘𝐵)∃ℎ ∈ (1^st ‘𝐶)𝑡 = (𝑔𝐺ℎ) ∧ 𝑥 = (𝑓𝐺𝑡))))
32		genpelvl.2	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝑦 ∈ Q ∧ 𝑧 ∈ Q) → (𝑦𝐺𝑧) ∈ Q)
33	32	caovcl 5987	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝑔 ∈ Q ∧ ℎ ∈ Q) → (𝑔𝐺ℎ) ∈ Q)
34		elisset 2735	. . . . . . . . . . . . . . . . . . . . . . 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 1889	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (∃𝑡(𝑡 = (𝑔𝐺ℎ) ∧ 𝑥 = ((𝑓𝐺𝑔)𝐺ℎ)) ↔ (∃𝑡 𝑡 = (𝑔𝐺ℎ) ∧ 𝑥 = ((𝑓𝐺𝑔)𝐺ℎ)))
38	36, 37	bitr4di 197	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑔 ∈ Q ∧ ℎ ∈ Q) → (𝑥 = ((𝑓𝐺𝑔)𝐺ℎ) ↔ ∃𝑡(𝑡 = (𝑔𝐺ℎ) ∧ 𝑥 = ((𝑓𝐺𝑔)𝐺ℎ))))
39	9, 38	sylan2 284	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑔 ∈ Q ∧ (𝐶 ∈ P ∧ ℎ ∈ (1^st ‘𝐶))) → (𝑥 = ((𝑓𝐺𝑔)𝐺ℎ) ↔ ∃𝑡(𝑡 = (𝑔𝐺ℎ) ∧ 𝑥 = ((𝑓𝐺𝑔)𝐺ℎ))))
40	39	anassrs 398	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝑔 ∈ Q ∧ 𝐶 ∈ P) ∧ ℎ ∈ (1^st ‘𝐶)) → (𝑥 = ((𝑓𝐺𝑔)𝐺ℎ) ↔ ∃𝑡(𝑡 = (𝑔𝐺ℎ) ∧ 𝑥 = ((𝑓𝐺𝑔)𝐺ℎ))))
41	40	rexbidva 2461	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑔 ∈ Q ∧ 𝐶 ∈ P) → (∃ℎ ∈ (1^st ‘𝐶)𝑥 = ((𝑓𝐺𝑔)𝐺ℎ) ↔ ∃ℎ ∈ (1^st ‘𝐶)∃𝑡(𝑡 = (𝑔𝐺ℎ) ∧ 𝑥 = ((𝑓𝐺𝑔)𝐺ℎ))))
42		rexcom4 2744	. . . . . . . . . . . . . . . . 17 ⊢ (∃ℎ ∈ (1^st ‘𝐶)∃𝑡(𝑡 = (𝑔𝐺ℎ) ∧ 𝑥 = ((𝑓𝐺𝑔)𝐺ℎ)) ↔ ∃𝑡∃ℎ ∈ (1^st ‘𝐶)(𝑡 = (𝑔𝐺ℎ) ∧ 𝑥 = ((𝑓𝐺𝑔)𝐺ℎ)))
43	41, 42	bitrdi 195	. . . . . . . . . . . . . . . 16 ⊢ ((𝑔 ∈ Q ∧ 𝐶 ∈ P) → (∃ℎ ∈ (1^st ‘𝐶)𝑥 = ((𝑓𝐺𝑔)𝐺ℎ) ↔ ∃𝑡∃ℎ ∈ (1^st ‘𝐶)(𝑡 = (𝑔𝐺ℎ) ∧ 𝑥 = ((𝑓𝐺𝑔)𝐺ℎ))))
44	43	ancoms 266	. . . . . . . . . . . . . . 15 ⊢ ((𝐶 ∈ P ∧ 𝑔 ∈ Q) → (∃ℎ ∈ (1^st ‘𝐶)𝑥 = ((𝑓𝐺𝑔)𝐺ℎ) ↔ ∃𝑡∃ℎ ∈ (1^st ‘𝐶)(𝑡 = (𝑔𝐺ℎ) ∧ 𝑥 = ((𝑓𝐺𝑔)𝐺ℎ))))
45	6, 44	sylan2 284	. . . . . . . . . . . . . 14 ⊢ ((𝐶 ∈ P ∧ (𝐵 ∈ P ∧ 𝑔 ∈ (1^st ‘𝐵))) → (∃ℎ ∈ (1^st ‘𝐶)𝑥 = ((𝑓𝐺𝑔)𝐺ℎ) ↔ ∃𝑡∃ℎ ∈ (1^st ‘𝐶)(𝑡 = (𝑔𝐺ℎ) ∧ 𝑥 = ((𝑓𝐺𝑔)𝐺ℎ))))
46	45	anassrs 398	. . . . . . . . . . . . 13 ⊢ (((𝐶 ∈ P ∧ 𝐵 ∈ P) ∧ 𝑔 ∈ (1^st ‘𝐵)) → (∃ℎ ∈ (1^st ‘𝐶)𝑥 = ((𝑓𝐺𝑔)𝐺ℎ) ↔ ∃𝑡∃ℎ ∈ (1^st ‘𝐶)(𝑡 = (𝑔𝐺ℎ) ∧ 𝑥 = ((𝑓𝐺𝑔)𝐺ℎ))))
47	46	rexbidva 2461	. . . . . . . . . . . 12 ⊢ ((𝐶 ∈ P ∧ 𝐵 ∈ P) → (∃𝑔 ∈ (1^st ‘𝐵)∃ℎ ∈ (1^st ‘𝐶)𝑥 = ((𝑓𝐺𝑔)𝐺ℎ) ↔ ∃𝑔 ∈ (1^st ‘𝐵)∃𝑡∃ℎ ∈ (1^st ‘𝐶)(𝑡 = (𝑔𝐺ℎ) ∧ 𝑥 = ((𝑓𝐺𝑔)𝐺ℎ))))
48	47	ancoms 266	. . . . . . . . . . 11 ⊢ ((𝐵 ∈ P ∧ 𝐶 ∈ P) → (∃𝑔 ∈ (1^st ‘𝐵)∃ℎ ∈ (1^st ‘𝐶)𝑥 = ((𝑓𝐺𝑔)𝐺ℎ) ↔ ∃𝑔 ∈ (1^st ‘𝐵)∃𝑡∃ℎ ∈ (1^st ‘𝐶)(𝑡 = (𝑔𝐺ℎ) ∧ 𝑥 = ((𝑓𝐺𝑔)𝐺ℎ))))
49		rexcom4 2744	. . . . . . . . . . 11 ⊢ (∃𝑔 ∈ (1^st ‘𝐵)∃𝑡∃ℎ ∈ (1^st ‘𝐶)(𝑡 = (𝑔𝐺ℎ) ∧ 𝑥 = ((𝑓𝐺𝑔)𝐺ℎ)) ↔ ∃𝑡∃𝑔 ∈ (1^st ‘𝐵)∃ℎ ∈ (1^st ‘𝐶)(𝑡 = (𝑔𝐺ℎ) ∧ 𝑥 = ((𝑓𝐺𝑔)𝐺ℎ)))
50	48, 49	bitrdi 195	. . . . . . . . . 10 ⊢ ((𝐵 ∈ P ∧ 𝐶 ∈ P) → (∃𝑔 ∈ (1^st ‘𝐵)∃ℎ ∈ (1^st ‘𝐶)𝑥 = ((𝑓𝐺𝑔)𝐺ℎ) ↔ ∃𝑡∃𝑔 ∈ (1^st ‘𝐵)∃ℎ ∈ (1^st ‘𝐶)(𝑡 = (𝑔𝐺ℎ) ∧ 𝑥 = ((𝑓𝐺𝑔)𝐺ℎ))))
51	50	adantr 274	. . . . . . . . 9 ⊢ (((𝐵 ∈ P ∧ 𝐶 ∈ P) ∧ 𝑓 ∈ Q) → (∃𝑔 ∈ (1^st ‘𝐵)∃ℎ ∈ (1^st ‘𝐶)𝑥 = ((𝑓𝐺𝑔)𝐺ℎ) ↔ ∃𝑡∃𝑔 ∈ (1^st ‘𝐵)∃ℎ ∈ (1^st ‘𝐶)(𝑡 = (𝑔𝐺ℎ) ∧ 𝑥 = ((𝑓𝐺𝑔)𝐺ℎ))))
52		df-rex 2448	. . . . . . . . . . 11 ⊢ (∃𝑡 ∈ (1^st ‘(𝐵𝐹𝐶))𝑥 = (𝑓𝐺𝑡) ↔ ∃𝑡(𝑡 ∈ (1^st ‘(𝐵𝐹𝐶)) ∧ 𝑥 = (𝑓𝐺𝑡)))
53		genpelvl.1	. . . . . . . . . . . . . 14 ⊢ 𝐹 = (𝑤 ∈ P, 𝑣 ∈ P ↦ ⟨{𝑥 ∈ Q ∣ ∃𝑦 ∈ Q ∃𝑧 ∈ Q (𝑦 ∈ (1^st ‘𝑤) ∧ 𝑧 ∈ (1^st ‘𝑣) ∧ 𝑥 = (𝑦𝐺𝑧))}, {𝑥 ∈ Q ∣ ∃𝑦 ∈ Q ∃𝑧 ∈ Q (𝑦 ∈ (2^nd ‘𝑤) ∧ 𝑧 ∈ (2^nd ‘𝑣) ∧ 𝑥 = (𝑦𝐺𝑧))}⟩)
54	53, 32	genpelvl 7444	. . . . . . . . . . . . 13 ⊢ ((𝐵 ∈ P ∧ 𝐶 ∈ P) → (𝑡 ∈ (1^st ‘(𝐵𝐹𝐶)) ↔ ∃𝑔 ∈ (1^st ‘𝐵)∃ℎ ∈ (1^st ‘𝐶)𝑡 = (𝑔𝐺ℎ)))
55	54	anbi1d 461	. . . . . . . . . . . 12 ⊢ ((𝐵 ∈ P ∧ 𝐶 ∈ P) → ((𝑡 ∈ (1^st ‘(𝐵𝐹𝐶)) ∧ 𝑥 = (𝑓𝐺𝑡)) ↔ (∃𝑔 ∈ (1^st ‘𝐵)∃ℎ ∈ (1^st ‘𝐶)𝑡 = (𝑔𝐺ℎ) ∧ 𝑥 = (𝑓𝐺𝑡))))
56	55	exbidv 1812	. . . . . . . . . . 11 ⊢ ((𝐵 ∈ P ∧ 𝐶 ∈ P) → (∃𝑡(𝑡 ∈ (1^st ‘(𝐵𝐹𝐶)) ∧ 𝑥 = (𝑓𝐺𝑡)) ↔ ∃𝑡(∃𝑔 ∈ (1^st ‘𝐵)∃ℎ ∈ (1^st ‘𝐶)𝑡 = (𝑔𝐺ℎ) ∧ 𝑥 = (𝑓𝐺𝑡))))
57	52, 56	syl5bb 191	. . . . . . . . . 10 ⊢ ((𝐵 ∈ P ∧ 𝐶 ∈ P) → (∃𝑡 ∈ (1^st ‘(𝐵𝐹𝐶))𝑥 = (𝑓𝐺𝑡) ↔ ∃𝑡(∃𝑔 ∈ (1^st ‘𝐵)∃ℎ ∈ (1^st ‘𝐶)𝑡 = (𝑔𝐺ℎ) ∧ 𝑥 = (𝑓𝐺𝑡))))
58	57	adantr 274	. . . . . . . . 9 ⊢ (((𝐵 ∈ P ∧ 𝐶 ∈ P) ∧ 𝑓 ∈ Q) → (∃𝑡 ∈ (1^st ‘(𝐵𝐹𝐶))𝑥 = (𝑓𝐺𝑡) ↔ ∃𝑡(∃𝑔 ∈ (1^st ‘𝐵)∃ℎ ∈ (1^st ‘𝐶)𝑡 = (𝑔𝐺ℎ) ∧ 𝑥 = (𝑓𝐺𝑡))))
59	31, 51, 58	3bitr4rd 220	. . . . . . . 8 ⊢ (((𝐵 ∈ P ∧ 𝐶 ∈ P) ∧ 𝑓 ∈ Q) → (∃𝑡 ∈ (1^st ‘(𝐵𝐹𝐶))𝑥 = (𝑓𝐺𝑡) ↔ ∃𝑔 ∈ (1^st ‘𝐵)∃ℎ ∈ (1^st ‘𝐶)𝑥 = ((𝑓𝐺𝑔)𝐺ℎ)))
60	3, 59	sylan2 284	. . . . . . 7 ⊢ (((𝐵 ∈ P ∧ 𝐶 ∈ P) ∧ (𝐴 ∈ P ∧ 𝑓 ∈ (1^st ‘𝐴))) → (∃𝑡 ∈ (1^st ‘(𝐵𝐹𝐶))𝑥 = (𝑓𝐺𝑡) ↔ ∃𝑔 ∈ (1^st ‘𝐵)∃ℎ ∈ (1^st ‘𝐶)𝑥 = ((𝑓𝐺𝑔)𝐺ℎ)))
61	60	anassrs 398	. . . . . 6 ⊢ ((((𝐵 ∈ P ∧ 𝐶 ∈ P) ∧ 𝐴 ∈ P) ∧ 𝑓 ∈ (1^st ‘𝐴)) → (∃𝑡 ∈ (1^st ‘(𝐵𝐹𝐶))𝑥 = (𝑓𝐺𝑡) ↔ ∃𝑔 ∈ (1^st ‘𝐵)∃ℎ ∈ (1^st ‘𝐶)𝑥 = ((𝑓𝐺𝑔)𝐺ℎ)))
62	61	rexbidva 2461	. . . . 5 ⊢ (((𝐵 ∈ P ∧ 𝐶 ∈ P) ∧ 𝐴 ∈ P) → (∃𝑓 ∈ (1^st ‘𝐴)∃𝑡 ∈ (1^st ‘(𝐵𝐹𝐶))𝑥 = (𝑓𝐺𝑡) ↔ ∃𝑓 ∈ (1^st ‘𝐴)∃𝑔 ∈ (1^st ‘𝐵)∃ℎ ∈ (1^st ‘𝐶)𝑥 = ((𝑓𝐺𝑔)𝐺ℎ)))
63	62	ancoms 266	. . . 4 ⊢ ((𝐴 ∈ P ∧ (𝐵 ∈ P ∧ 𝐶 ∈ P)) → (∃𝑓 ∈ (1^st ‘𝐴)∃𝑡 ∈ (1^st ‘(𝐵𝐹𝐶))𝑥 = (𝑓𝐺𝑡) ↔ ∃𝑓 ∈ (1^st ‘𝐴)∃𝑔 ∈ (1^st ‘𝐵)∃ℎ ∈ (1^st ‘𝐶)𝑥 = ((𝑓𝐺𝑔)𝐺ℎ)))
64	63	3impb 1188	. . 3 ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P ∧ 𝐶 ∈ P) → (∃𝑓 ∈ (1^st ‘𝐴)∃𝑡 ∈ (1^st ‘(𝐵𝐹𝐶))𝑥 = (𝑓𝐺𝑡) ↔ ∃𝑓 ∈ (1^st ‘𝐴)∃𝑔 ∈ (1^st ‘𝐵)∃ℎ ∈ (1^st ‘𝐶)𝑥 = ((𝑓𝐺𝑔)𝐺ℎ)))
65		genpassg.5	. . . . . 6 ⊢ ((𝑓 ∈ P ∧ 𝑔 ∈ P) → (𝑓𝐹𝑔) ∈ P)
66	65	caovcl 5987	. . . . 5 ⊢ ((𝐵 ∈ P ∧ 𝐶 ∈ P) → (𝐵𝐹𝐶) ∈ P)
67	53, 32	genpelvl 7444	. . . . 5 ⊢ ((𝐴 ∈ P ∧ (𝐵𝐹𝐶) ∈ P) → (𝑥 ∈ (1^st ‘(𝐴𝐹(𝐵𝐹𝐶))) ↔ ∃𝑓 ∈ (1^st ‘𝐴)∃𝑡 ∈ (1^st ‘(𝐵𝐹𝐶))𝑥 = (𝑓𝐺𝑡)))
68	66, 67	sylan2 284	. . . 4 ⊢ ((𝐴 ∈ P ∧ (𝐵 ∈ P ∧ 𝐶 ∈ P)) → (𝑥 ∈ (1^st ‘(𝐴𝐹(𝐵𝐹𝐶))) ↔ ∃𝑓 ∈ (1^st ‘𝐴)∃𝑡 ∈ (1^st ‘(𝐵𝐹𝐶))𝑥 = (𝑓𝐺𝑡)))
69	68	3impb 1188	. . 3 ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P ∧ 𝐶 ∈ P) → (𝑥 ∈ (1^st ‘(𝐴𝐹(𝐵𝐹𝐶))) ↔ ∃𝑓 ∈ (1^st ‘𝐴)∃𝑡 ∈ (1^st ‘(𝐵𝐹𝐶))𝑥 = (𝑓𝐺𝑡)))
70		df-rex 2448	. . . . 5 ⊢ (∃𝑡 ∈ (1^st ‘(𝐴𝐹𝐵))∃ℎ ∈ (1^st ‘𝐶)𝑥 = (𝑡𝐺ℎ) ↔ ∃𝑡(𝑡 ∈ (1^st ‘(𝐴𝐹𝐵)) ∧ ∃ℎ ∈ (1^st ‘𝐶)𝑥 = (𝑡𝐺ℎ)))
71	53, 32	genpelvl 7444	. . . . . . . 8 ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P) → (𝑡 ∈ (1^st ‘(𝐴𝐹𝐵)) ↔ ∃𝑓 ∈ (1^st ‘𝐴)∃𝑔 ∈ (1^st ‘𝐵)𝑡 = (𝑓𝐺𝑔)))
72	71	3adant3 1006	. . . . . . 7 ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P ∧ 𝐶 ∈ P) → (𝑡 ∈ (1^st ‘(𝐴𝐹𝐵)) ↔ ∃𝑓 ∈ (1^st ‘𝐴)∃𝑔 ∈ (1^st ‘𝐵)𝑡 = (𝑓𝐺𝑔)))
73	72	anbi1d 461	. . . . . 6 ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P ∧ 𝐶 ∈ P) → ((𝑡 ∈ (1^st ‘(𝐴𝐹𝐵)) ∧ ∃ℎ ∈ (1^st ‘𝐶)𝑥 = (𝑡𝐺ℎ)) ↔ (∃𝑓 ∈ (1^st ‘𝐴)∃𝑔 ∈ (1^st ‘𝐵)𝑡 = (𝑓𝐺𝑔) ∧ ∃ℎ ∈ (1^st ‘𝐶)𝑥 = (𝑡𝐺ℎ))))
74	73	exbidv 1812	. . . . 5 ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P ∧ 𝐶 ∈ P) → (∃𝑡(𝑡 ∈ (1^st ‘(𝐴𝐹𝐵)) ∧ ∃ℎ ∈ (1^st ‘𝐶)𝑥 = (𝑡𝐺ℎ)) ↔ ∃𝑡(∃𝑓 ∈ (1^st ‘𝐴)∃𝑔 ∈ (1^st ‘𝐵)𝑡 = (𝑓𝐺𝑔) ∧ ∃ℎ ∈ (1^st ‘𝐶)𝑥 = (𝑡𝐺ℎ))))
75	70, 74	syl5bb 191	. . . 4 ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P ∧ 𝐶 ∈ P) → (∃𝑡 ∈ (1^st ‘(𝐴𝐹𝐵))∃ℎ ∈ (1^st ‘𝐶)𝑥 = (𝑡𝐺ℎ) ↔ ∃𝑡(∃𝑓 ∈ (1^st ‘𝐴)∃𝑔 ∈ (1^st ‘𝐵)𝑡 = (𝑓𝐺𝑔) ∧ ∃ℎ ∈ (1^st ‘𝐶)𝑥 = (𝑡𝐺ℎ))))
76	65	caovcl 5987	. . . . . 6 ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P) → (𝐴𝐹𝐵) ∈ P)
77	53, 32	genpelvl 7444	. . . . . 6 ⊢ (((𝐴𝐹𝐵) ∈ P ∧ 𝐶 ∈ P) → (𝑥 ∈ (1^st ‘((𝐴𝐹𝐵)𝐹𝐶)) ↔ ∃𝑡 ∈ (1^st ‘(𝐴𝐹𝐵))∃ℎ ∈ (1^st ‘𝐶)𝑥 = (𝑡𝐺ℎ)))
78	76, 77	sylan 281	. . . . 5 ⊢ (((𝐴 ∈ P ∧ 𝐵 ∈ P) ∧ 𝐶 ∈ P) → (𝑥 ∈ (1^st ‘((𝐴𝐹𝐵)𝐹𝐶)) ↔ ∃𝑡 ∈ (1^st ‘(𝐴𝐹𝐵))∃ℎ ∈ (1^st ‘𝐶)𝑥 = (𝑡𝐺ℎ)))
79	78	3impa 1183	. . . 4 ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P ∧ 𝐶 ∈ P) → (𝑥 ∈ (1^st ‘((𝐴𝐹𝐵)𝐹𝐶)) ↔ ∃𝑡 ∈ (1^st ‘(𝐴𝐹𝐵))∃ℎ ∈ (1^st ‘𝐶)𝑥 = (𝑡𝐺ℎ)))
80	32	caovcl 5987	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑓 ∈ Q ∧ 𝑔 ∈ Q) → (𝑓𝐺𝑔) ∈ Q)
81		elisset 2735	. . . . . . . . . . . . . . . . . . 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 ∈ (1^st ‘𝐶)𝑥 = ((𝑓𝐺𝑔)𝐺ℎ) ↔ (∃𝑡 𝑡 = (𝑓𝐺𝑔) ∧ ∃ℎ ∈ (1^st ‘𝐶)𝑥 = ((𝑓𝐺𝑔)𝐺ℎ))))
84		oveq1 5843	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑡 = (𝑓𝐺𝑔) → (𝑡𝐺ℎ) = ((𝑓𝐺𝑔)𝐺ℎ))
85	84	eqeq2d 2176	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑡 = (𝑓𝐺𝑔) → (𝑥 = (𝑡𝐺ℎ) ↔ 𝑥 = ((𝑓𝐺𝑔)𝐺ℎ)))
86	85	rexbidv 2465	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑡 = (𝑓𝐺𝑔) → (∃ℎ ∈ (1^st ‘𝐶)𝑥 = (𝑡𝐺ℎ) ↔ ∃ℎ ∈ (1^st ‘𝐶)𝑥 = ((𝑓𝐺𝑔)𝐺ℎ)))
87	86	pm5.32i 450	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑡 = (𝑓𝐺𝑔) ∧ ∃ℎ ∈ (1^st ‘𝐶)𝑥 = (𝑡𝐺ℎ)) ↔ (𝑡 = (𝑓𝐺𝑔) ∧ ∃ℎ ∈ (1^st ‘𝐶)𝑥 = ((𝑓𝐺𝑔)𝐺ℎ)))
88	87	exbii 1592	. . . . . . . . . . . . . . . . . 18 ⊢ (∃𝑡(𝑡 = (𝑓𝐺𝑔) ∧ ∃ℎ ∈ (1^st ‘𝐶)𝑥 = (𝑡𝐺ℎ)) ↔ ∃𝑡(𝑡 = (𝑓𝐺𝑔) ∧ ∃ℎ ∈ (1^st ‘𝐶)𝑥 = ((𝑓𝐺𝑔)𝐺ℎ)))
89		19.41v 1889	. . . . . . . . . . . . . . . . . 18 ⊢ (∃𝑡(𝑡 = (𝑓𝐺𝑔) ∧ ∃ℎ ∈ (1^st ‘𝐶)𝑥 = ((𝑓𝐺𝑔)𝐺ℎ)) ↔ (∃𝑡 𝑡 = (𝑓𝐺𝑔) ∧ ∃ℎ ∈ (1^st ‘𝐶)𝑥 = ((𝑓𝐺𝑔)𝐺ℎ)))
90	88, 89	bitri 183	. . . . . . . . . . . . . . . . 17 ⊢ (∃𝑡(𝑡 = (𝑓𝐺𝑔) ∧ ∃ℎ ∈ (1^st ‘𝐶)𝑥 = (𝑡𝐺ℎ)) ↔ (∃𝑡 𝑡 = (𝑓𝐺𝑔) ∧ ∃ℎ ∈ (1^st ‘𝐶)𝑥 = ((𝑓𝐺𝑔)𝐺ℎ)))
91	83, 90	bitr4di 197	. . . . . . . . . . . . . . . 16 ⊢ ((𝑓 ∈ Q ∧ 𝑔 ∈ Q) → (∃ℎ ∈ (1^st ‘𝐶)𝑥 = ((𝑓𝐺𝑔)𝐺ℎ) ↔ ∃𝑡(𝑡 = (𝑓𝐺𝑔) ∧ ∃ℎ ∈ (1^st ‘𝐶)𝑥 = (𝑡𝐺ℎ))))
92	6, 91	sylan2 284	. . . . . . . . . . . . . . 15 ⊢ ((𝑓 ∈ Q ∧ (𝐵 ∈ P ∧ 𝑔 ∈ (1^st ‘𝐵))) → (∃ℎ ∈ (1^st ‘𝐶)𝑥 = ((𝑓𝐺𝑔)𝐺ℎ) ↔ ∃𝑡(𝑡 = (𝑓𝐺𝑔) ∧ ∃ℎ ∈ (1^st ‘𝐶)𝑥 = (𝑡𝐺ℎ))))
93	92	anassrs 398	. . . . . . . . . . . . . 14 ⊢ (((𝑓 ∈ Q ∧ 𝐵 ∈ P) ∧ 𝑔 ∈ (1^st ‘𝐵)) → (∃ℎ ∈ (1^st ‘𝐶)𝑥 = ((𝑓𝐺𝑔)𝐺ℎ) ↔ ∃𝑡(𝑡 = (𝑓𝐺𝑔) ∧ ∃ℎ ∈ (1^st ‘𝐶)𝑥 = (𝑡𝐺ℎ))))
94	93	rexbidva 2461	. . . . . . . . . . . . 13 ⊢ ((𝑓 ∈ Q ∧ 𝐵 ∈ P) → (∃𝑔 ∈ (1^st ‘𝐵)∃ℎ ∈ (1^st ‘𝐶)𝑥 = ((𝑓𝐺𝑔)𝐺ℎ) ↔ ∃𝑔 ∈ (1^st ‘𝐵)∃𝑡(𝑡 = (𝑓𝐺𝑔) ∧ ∃ℎ ∈ (1^st ‘𝐶)𝑥 = (𝑡𝐺ℎ))))
95		rexcom4 2744	. . . . . . . . . . . . 13 ⊢ (∃𝑔 ∈ (1^st ‘𝐵)∃𝑡(𝑡 = (𝑓𝐺𝑔) ∧ ∃ℎ ∈ (1^st ‘𝐶)𝑥 = (𝑡𝐺ℎ)) ↔ ∃𝑡∃𝑔 ∈ (1^st ‘𝐵)(𝑡 = (𝑓𝐺𝑔) ∧ ∃ℎ ∈ (1^st ‘𝐶)𝑥 = (𝑡𝐺ℎ)))
96	94, 95	bitrdi 195	. . . . . . . . . . . 12 ⊢ ((𝑓 ∈ Q ∧ 𝐵 ∈ P) → (∃𝑔 ∈ (1^st ‘𝐵)∃ℎ ∈ (1^st ‘𝐶)𝑥 = ((𝑓𝐺𝑔)𝐺ℎ) ↔ ∃𝑡∃𝑔 ∈ (1^st ‘𝐵)(𝑡 = (𝑓𝐺𝑔) ∧ ∃ℎ ∈ (1^st ‘𝐶)𝑥 = (𝑡𝐺ℎ))))
97	96	ancoms 266	. . . . . . . . . . 11 ⊢ ((𝐵 ∈ P ∧ 𝑓 ∈ Q) → (∃𝑔 ∈ (1^st ‘𝐵)∃ℎ ∈ (1^st ‘𝐶)𝑥 = ((𝑓𝐺𝑔)𝐺ℎ) ↔ ∃𝑡∃𝑔 ∈ (1^st ‘𝐵)(𝑡 = (𝑓𝐺𝑔) ∧ ∃ℎ ∈ (1^st ‘𝐶)𝑥 = (𝑡𝐺ℎ))))
98	3, 97	sylan2 284	. . . . . . . . . 10 ⊢ ((𝐵 ∈ P ∧ (𝐴 ∈ P ∧ 𝑓 ∈ (1^st ‘𝐴))) → (∃𝑔 ∈ (1^st ‘𝐵)∃ℎ ∈ (1^st ‘𝐶)𝑥 = ((𝑓𝐺𝑔)𝐺ℎ) ↔ ∃𝑡∃𝑔 ∈ (1^st ‘𝐵)(𝑡 = (𝑓𝐺𝑔) ∧ ∃ℎ ∈ (1^st ‘𝐶)𝑥 = (𝑡𝐺ℎ))))
99	98	anassrs 398	. . . . . . . . 9 ⊢ (((𝐵 ∈ P ∧ 𝐴 ∈ P) ∧ 𝑓 ∈ (1^st ‘𝐴)) → (∃𝑔 ∈ (1^st ‘𝐵)∃ℎ ∈ (1^st ‘𝐶)𝑥 = ((𝑓𝐺𝑔)𝐺ℎ) ↔ ∃𝑡∃𝑔 ∈ (1^st ‘𝐵)(𝑡 = (𝑓𝐺𝑔) ∧ ∃ℎ ∈ (1^st ‘𝐶)𝑥 = (𝑡𝐺ℎ))))
100	99	rexbidva 2461	. . . . . . . 8 ⊢ ((𝐵 ∈ P ∧ 𝐴 ∈ P) → (∃𝑓 ∈ (1^st ‘𝐴)∃𝑔 ∈ (1^st ‘𝐵)∃ℎ ∈ (1^st ‘𝐶)𝑥 = ((𝑓𝐺𝑔)𝐺ℎ) ↔ ∃𝑓 ∈ (1^st ‘𝐴)∃𝑡∃𝑔 ∈ (1^st ‘𝐵)(𝑡 = (𝑓𝐺𝑔) ∧ ∃ℎ ∈ (1^st ‘𝐶)𝑥 = (𝑡𝐺ℎ))))
101		rexcom4 2744	. . . . . . . 8 ⊢ (∃𝑓 ∈ (1^st ‘𝐴)∃𝑡∃𝑔 ∈ (1^st ‘𝐵)(𝑡 = (𝑓𝐺𝑔) ∧ ∃ℎ ∈ (1^st ‘𝐶)𝑥 = (𝑡𝐺ℎ)) ↔ ∃𝑡∃𝑓 ∈ (1^st ‘𝐴)∃𝑔 ∈ (1^st ‘𝐵)(𝑡 = (𝑓𝐺𝑔) ∧ ∃ℎ ∈ (1^st ‘𝐶)𝑥 = (𝑡𝐺ℎ)))
102	100, 101	bitrdi 195	. . . . . . 7 ⊢ ((𝐵 ∈ P ∧ 𝐴 ∈ P) → (∃𝑓 ∈ (1^st ‘𝐴)∃𝑔 ∈ (1^st ‘𝐵)∃ℎ ∈ (1^st ‘𝐶)𝑥 = ((𝑓𝐺𝑔)𝐺ℎ) ↔ ∃𝑡∃𝑓 ∈ (1^st ‘𝐴)∃𝑔 ∈ (1^st ‘𝐵)(𝑡 = (𝑓𝐺𝑔) ∧ ∃ℎ ∈ (1^st ‘𝐶)𝑥 = (𝑡𝐺ℎ))))
103		r19.41v 2620	. . . . . . . . . 10 ⊢ (∃𝑔 ∈ (1^st ‘𝐵)(𝑡 = (𝑓𝐺𝑔) ∧ ∃ℎ ∈ (1^st ‘𝐶)𝑥 = (𝑡𝐺ℎ)) ↔ (∃𝑔 ∈ (1^st ‘𝐵)𝑡 = (𝑓𝐺𝑔) ∧ ∃ℎ ∈ (1^st ‘𝐶)𝑥 = (𝑡𝐺ℎ)))
104	103	rexbii 2471	. . . . . . . . 9 ⊢ (∃𝑓 ∈ (1^st ‘𝐴)∃𝑔 ∈ (1^st ‘𝐵)(𝑡 = (𝑓𝐺𝑔) ∧ ∃ℎ ∈ (1^st ‘𝐶)𝑥 = (𝑡𝐺ℎ)) ↔ ∃𝑓 ∈ (1^st ‘𝐴)(∃𝑔 ∈ (1^st ‘𝐵)𝑡 = (𝑓𝐺𝑔) ∧ ∃ℎ ∈ (1^st ‘𝐶)𝑥 = (𝑡𝐺ℎ)))
105		r19.41v 2620	. . . . . . . . 9 ⊢ (∃𝑓 ∈ (1^st ‘𝐴)(∃𝑔 ∈ (1^st ‘𝐵)𝑡 = (𝑓𝐺𝑔) ∧ ∃ℎ ∈ (1^st ‘𝐶)𝑥 = (𝑡𝐺ℎ)) ↔ (∃𝑓 ∈ (1^st ‘𝐴)∃𝑔 ∈ (1^st ‘𝐵)𝑡 = (𝑓𝐺𝑔) ∧ ∃ℎ ∈ (1^st ‘𝐶)𝑥 = (𝑡𝐺ℎ)))
106	104, 105	bitri 183	. . . . . . . 8 ⊢ (∃𝑓 ∈ (1^st ‘𝐴)∃𝑔 ∈ (1^st ‘𝐵)(𝑡 = (𝑓𝐺𝑔) ∧ ∃ℎ ∈ (1^st ‘𝐶)𝑥 = (𝑡𝐺ℎ)) ↔ (∃𝑓 ∈ (1^st ‘𝐴)∃𝑔 ∈ (1^st ‘𝐵)𝑡 = (𝑓𝐺𝑔) ∧ ∃ℎ ∈ (1^st ‘𝐶)𝑥 = (𝑡𝐺ℎ)))
107	106	exbii 1592	. . . . . . 7 ⊢ (∃𝑡∃𝑓 ∈ (1^st ‘𝐴)∃𝑔 ∈ (1^st ‘𝐵)(𝑡 = (𝑓𝐺𝑔) ∧ ∃ℎ ∈ (1^st ‘𝐶)𝑥 = (𝑡𝐺ℎ)) ↔ ∃𝑡(∃𝑓 ∈ (1^st ‘𝐴)∃𝑔 ∈ (1^st ‘𝐵)𝑡 = (𝑓𝐺𝑔) ∧ ∃ℎ ∈ (1^st ‘𝐶)𝑥 = (𝑡𝐺ℎ)))
108	102, 107	bitrdi 195	. . . . . 6 ⊢ ((𝐵 ∈ P ∧ 𝐴 ∈ P) → (∃𝑓 ∈ (1^st ‘𝐴)∃𝑔 ∈ (1^st ‘𝐵)∃ℎ ∈ (1^st ‘𝐶)𝑥 = ((𝑓𝐺𝑔)𝐺ℎ) ↔ ∃𝑡(∃𝑓 ∈ (1^st ‘𝐴)∃𝑔 ∈ (1^st ‘𝐵)𝑡 = (𝑓𝐺𝑔) ∧ ∃ℎ ∈ (1^st ‘𝐶)𝑥 = (𝑡𝐺ℎ))))
109	108	ancoms 266	. . . . 5 ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P) → (∃𝑓 ∈ (1^st ‘𝐴)∃𝑔 ∈ (1^st ‘𝐵)∃ℎ ∈ (1^st ‘𝐶)𝑥 = ((𝑓𝐺𝑔)𝐺ℎ) ↔ ∃𝑡(∃𝑓 ∈ (1^st ‘𝐴)∃𝑔 ∈ (1^st ‘𝐵)𝑡 = (𝑓𝐺𝑔) ∧ ∃ℎ ∈ (1^st ‘𝐶)𝑥 = (𝑡𝐺ℎ))))
110	109	3adant3 1006	. . . 4 ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P ∧ 𝐶 ∈ P) → (∃𝑓 ∈ (1^st ‘𝐴)∃𝑔 ∈ (1^st ‘𝐵)∃ℎ ∈ (1^st ‘𝐶)𝑥 = ((𝑓𝐺𝑔)𝐺ℎ) ↔ ∃𝑡(∃𝑓 ∈ (1^st ‘𝐴)∃𝑔 ∈ (1^st ‘𝐵)𝑡 = (𝑓𝐺𝑔) ∧ ∃ℎ ∈ (1^st ‘𝐶)𝑥 = (𝑡𝐺ℎ))))
111	75, 79, 110	3bitr4d 219	. . 3 ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P ∧ 𝐶 ∈ P) → (𝑥 ∈ (1^st ‘((𝐴𝐹𝐵)𝐹𝐶)) ↔ ∃𝑓 ∈ (1^st ‘𝐴)∃𝑔 ∈ (1^st ‘𝐵)∃ℎ ∈ (1^st ‘𝐶)𝑥 = ((𝑓𝐺𝑔)𝐺ℎ)))
112	64, 69, 111	3bitr4rd 220	. 2 ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P ∧ 𝐶 ∈ P) → (𝑥 ∈ (1^st ‘((𝐴𝐹𝐵)𝐹𝐶)) ↔ 𝑥 ∈ (1^st ‘(𝐴𝐹(𝐵𝐹𝐶)))))
113	112	eqrdv 2162	1 ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P ∧ 𝐶 ∈ P) → (1^st ‘((𝐴𝐹𝐵)𝐹𝐶)) = (1^st ‘(𝐴𝐹(𝐵𝐹𝐶))))

Colors of variables: wff set class

Syntax hints: → wi 4 ∧ wa 103 ↔ wb 104 ∧ w3a 967 = wceq 1342 ∃wex 1479 ∈ wcel 2135 ∃wrex 2443 {crab 2446 ⟨cop 3573 × cxp 4596 dom cdm 4598 ‘cfv 5182 (class class class)co 5836 ∈ cmpo 5838 1^st c1st 6098 2^nd c2nd 6099 Qcnq 7212 Pcnp 7223

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 604 ax-in2 605 ax-io 699 ax-5 1434 ax-7 1435 ax-gen 1436 ax-ie1 1480 ax-ie2 1481 ax-8 1491 ax-10 1492 ax-11 1493 ax-i12 1494 ax-bndl 1496 ax-4 1497 ax-17 1513 ax-i9 1517 ax-ial 1521 ax-i5r 1522 ax-13 2137 ax-14 2138 ax-ext 2146 ax-coll 4091 ax-sep 4094 ax-pow 4147 ax-pr 4181 ax-un 4405 ax-setind 4508 ax-iinf 4559

This theorem depends on definitions: df-bi 116 df-3an 969 df-tru 1345 df-fal 1348 df-nf 1448 df-sb 1750 df-eu 2016 df-mo 2017 df-clab 2151 df-cleq 2157 df-clel 2160 df-nfc 2295 df-ne 2335 df-ral 2447 df-rex 2448 df-reu 2449 df-rab 2451 df-v 2723 df-sbc 2947 df-csb 3041 df-dif 3113 df-un 3115 df-in 3117 df-ss 3124 df-pw 3555 df-sn 3576 df-pr 3577 df-op 3579 df-uni 3784 df-int 3819 df-iun 3862 df-br 3977 df-opab 4038 df-mpt 4039 df-id 4265 df-iom 4562 df-xp 4604 df-rel 4605 df-cnv 4606 df-co 4607 df-dm 4608 df-rn 4609 df-res 4610 df-ima 4611 df-iota 5147 df-fun 5184 df-fn 5185 df-f 5186 df-f1 5187 df-fo 5188 df-f1o 5189 df-fv 5190 df-ov 5839 df-oprab 5840 df-mpo 5841 df-1st 6100 df-2nd 6101 df-qs 6498 df-ni 7236 df-nqqs 7280 df-inp 7398

This theorem is referenced by: genpassg 7458

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