Theorem genpass 10982

Description: Associativity of an operation on reals. (Contributed by NM, 18-Mar-1996.) (Revised by Mario Carneiro, 12-Jun-2013.) (New usage is discouraged.)

Hypotheses

Ref	Expression
genp.1	⊢ 𝐹 = (𝑤 ∈ P, 𝑣 ∈ P ↦ {𝑥 ∣ ∃𝑦 ∈ 𝑤 ∃𝑧 ∈ 𝑣 𝑥 = (𝑦𝐺𝑧)})
genp.2	⊢ ((𝑦 ∈ Q ∧ 𝑧 ∈ Q) → (𝑦𝐺𝑧) ∈ Q)
genpass.4	⊢ dom 𝐹 = (P × P)
genpass.5	⊢ ((𝑓 ∈ P ∧ 𝑔 ∈ P) → (𝑓𝐹𝑔) ∈ P)
genpass.6	⊢ ((𝑓𝐺𝑔)𝐺ℎ) = (𝑓𝐺(𝑔𝐺ℎ))

Assertion

Ref	Expression
genpass	⊢ ((𝐴𝐹𝐵)𝐹𝐶) = (𝐴𝐹(𝐵𝐹𝐶))

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

Allowed substitution hints:   𝐴(𝑤,𝑣)   𝐵(𝑤,𝑣)   𝐶(𝑤,𝑣)   𝐹(𝑤,𝑣)

Proof of Theorem genpass

Dummy variable 𝑡 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		genp.1	. . . . . . . . . 10 ⊢ 𝐹 = (𝑤 ∈ P, 𝑣 ∈ P ↦ {𝑥 ∣ ∃𝑦 ∈ 𝑤 ∃𝑧 ∈ 𝑣 𝑥 = (𝑦𝐺𝑧)})
2		genp.2	. . . . . . . . . 10 ⊢ ((𝑦 ∈ Q ∧ 𝑧 ∈ Q) → (𝑦𝐺𝑧) ∈ Q)
3	1, 2	genpelv 10973	. . . . . . . . 9 ⊢ ((𝐵 ∈ P ∧ 𝐶 ∈ P) → (𝑡 ∈ (𝐵𝐹𝐶) ↔ ∃𝑔 ∈ 𝐵 ∃ℎ ∈ 𝐶 𝑡 = (𝑔𝐺ℎ)))
4	3	3adant1 1146	. . . . . . . 8 ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P ∧ 𝐶 ∈ P) → (𝑡 ∈ (𝐵𝐹𝐶) ↔ ∃𝑔 ∈ 𝐵 ∃ℎ ∈ 𝐶 𝑡 = (𝑔𝐺ℎ)))
5	4	anbi1d 642	. . . . . . 7 ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P ∧ 𝐶 ∈ P) → ((𝑡 ∈ (𝐵𝐹𝐶) ∧ 𝑥 = (𝑓𝐺𝑡)) ↔ (∃𝑔 ∈ 𝐵 ∃ℎ ∈ 𝐶 𝑡 = (𝑔𝐺ℎ) ∧ 𝑥 = (𝑓𝐺𝑡))))
6	5	exbidv 1944	. . . . . 6 ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P ∧ 𝐶 ∈ P) → (∃𝑡(𝑡 ∈ (𝐵𝐹𝐶) ∧ 𝑥 = (𝑓𝐺𝑡)) ↔ ∃𝑡(∃𝑔 ∈ 𝐵 ∃ℎ ∈ 𝐶 𝑡 = (𝑔𝐺ℎ) ∧ 𝑥 = (𝑓𝐺𝑡))))
7		df-rex 3090	. . . . . 6 ⊢ (∃𝑡 ∈ (𝐵𝐹𝐶)𝑥 = (𝑓𝐺𝑡) ↔ ∃𝑡(𝑡 ∈ (𝐵𝐹𝐶) ∧ 𝑥 = (𝑓𝐺𝑡)))
8		ovex 7433	. . . . . . . . . . . . 13 ⊢ (𝑔𝐺ℎ) ∈ V
9	8	isseti 3475	. . . . . . . . . . . 12 ⊢ ∃𝑡 𝑡 = (𝑔𝐺ℎ)
10	9	biantrur 539	. . . . . . . . . . 11 ⊢ (𝑥 = ((𝑓𝐺𝑔)𝐺ℎ) ↔ (∃𝑡 𝑡 = (𝑔𝐺ℎ) ∧ 𝑥 = ((𝑓𝐺𝑔)𝐺ℎ)))
11		19.41v 1972	. . . . . . . . . . 11 ⊢ (∃𝑡(𝑡 = (𝑔𝐺ℎ) ∧ 𝑥 = ((𝑓𝐺𝑔)𝐺ℎ)) ↔ (∃𝑡 𝑡 = (𝑔𝐺ℎ) ∧ 𝑥 = ((𝑓𝐺𝑔)𝐺ℎ)))
12	10, 11	bitr4i 281	. . . . . . . . . 10 ⊢ (𝑥 = ((𝑓𝐺𝑔)𝐺ℎ) ↔ ∃𝑡(𝑡 = (𝑔𝐺ℎ) ∧ 𝑥 = ((𝑓𝐺𝑔)𝐺ℎ)))
13	12	rexbii 3112	. . . . . . . . 9 ⊢ (∃ℎ ∈ 𝐶 𝑥 = ((𝑓𝐺𝑔)𝐺ℎ) ↔ ∃ℎ ∈ 𝐶 ∃𝑡(𝑡 = (𝑔𝐺ℎ) ∧ 𝑥 = ((𝑓𝐺𝑔)𝐺ℎ)))
14		rexcom4 3292	. . . . . . . . 9 ⊢ (∃ℎ ∈ 𝐶 ∃𝑡(𝑡 = (𝑔𝐺ℎ) ∧ 𝑥 = ((𝑓𝐺𝑔)𝐺ℎ)) ↔ ∃𝑡∃ℎ ∈ 𝐶 (𝑡 = (𝑔𝐺ℎ) ∧ 𝑥 = ((𝑓𝐺𝑔)𝐺ℎ)))
15	13, 14	bitri 278	. . . . . . . 8 ⊢ (∃ℎ ∈ 𝐶 𝑥 = ((𝑓𝐺𝑔)𝐺ℎ) ↔ ∃𝑡∃ℎ ∈ 𝐶 (𝑡 = (𝑔𝐺ℎ) ∧ 𝑥 = ((𝑓𝐺𝑔)𝐺ℎ)))
16	15	rexbii 3112	. . . . . . 7 ⊢ (∃𝑔 ∈ 𝐵 ∃ℎ ∈ 𝐶 𝑥 = ((𝑓𝐺𝑔)𝐺ℎ) ↔ ∃𝑔 ∈ 𝐵 ∃𝑡∃ℎ ∈ 𝐶 (𝑡 = (𝑔𝐺ℎ) ∧ 𝑥 = ((𝑓𝐺𝑔)𝐺ℎ)))
17		rexcom4 3292	. . . . . . 7 ⊢ (∃𝑔 ∈ 𝐵 ∃𝑡∃ℎ ∈ 𝐶 (𝑡 = (𝑔𝐺ℎ) ∧ 𝑥 = ((𝑓𝐺𝑔)𝐺ℎ)) ↔ ∃𝑡∃𝑔 ∈ 𝐵 ∃ℎ ∈ 𝐶 (𝑡 = (𝑔𝐺ℎ) ∧ 𝑥 = ((𝑓𝐺𝑔)𝐺ℎ)))
18		oveq2 7408	. . . . . . . . . . . . . . 15 ⊢ (𝑡 = (𝑔𝐺ℎ) → (𝑓𝐺𝑡) = (𝑓𝐺(𝑔𝐺ℎ)))
19		genpass.6	. . . . . . . . . . . . . . 15 ⊢ ((𝑓𝐺𝑔)𝐺ℎ) = (𝑓𝐺(𝑔𝐺ℎ))
20	18, 19	eqtr4di 2818	. . . . . . . . . . . . . 14 ⊢ (𝑡 = (𝑔𝐺ℎ) → (𝑓𝐺𝑡) = ((𝑓𝐺𝑔)𝐺ℎ))
21	20	eqeq2d 2776	. . . . . . . . . . . . 13 ⊢ (𝑡 = (𝑔𝐺ℎ) → (𝑥 = (𝑓𝐺𝑡) ↔ 𝑥 = ((𝑓𝐺𝑔)𝐺ℎ)))
22	21	pm5.32i 584	. . . . . . . . . . . 12 ⊢ ((𝑡 = (𝑔𝐺ℎ) ∧ 𝑥 = (𝑓𝐺𝑡)) ↔ (𝑡 = (𝑔𝐺ℎ) ∧ 𝑥 = ((𝑓𝐺𝑔)𝐺ℎ)))
23	22	rexbii 3112	. . . . . . . . . . 11 ⊢ (∃ℎ ∈ 𝐶 (𝑡 = (𝑔𝐺ℎ) ∧ 𝑥 = (𝑓𝐺𝑡)) ↔ ∃ℎ ∈ 𝐶 (𝑡 = (𝑔𝐺ℎ) ∧ 𝑥 = ((𝑓𝐺𝑔)𝐺ℎ)))
24		r19.41v 3195	. . . . . . . . . . 11 ⊢ (∃ℎ ∈ 𝐶 (𝑡 = (𝑔𝐺ℎ) ∧ 𝑥 = (𝑓𝐺𝑡)) ↔ (∃ℎ ∈ 𝐶 𝑡 = (𝑔𝐺ℎ) ∧ 𝑥 = (𝑓𝐺𝑡)))
25	23, 24	bitr3i 280	. . . . . . . . . 10 ⊢ (∃ℎ ∈ 𝐶 (𝑡 = (𝑔𝐺ℎ) ∧ 𝑥 = ((𝑓𝐺𝑔)𝐺ℎ)) ↔ (∃ℎ ∈ 𝐶 𝑡 = (𝑔𝐺ℎ) ∧ 𝑥 = (𝑓𝐺𝑡)))
26	25	rexbii 3112	. . . . . . . . 9 ⊢ (∃𝑔 ∈ 𝐵 ∃ℎ ∈ 𝐶 (𝑡 = (𝑔𝐺ℎ) ∧ 𝑥 = ((𝑓𝐺𝑔)𝐺ℎ)) ↔ ∃𝑔 ∈ 𝐵 (∃ℎ ∈ 𝐶 𝑡 = (𝑔𝐺ℎ) ∧ 𝑥 = (𝑓𝐺𝑡)))
27		r19.41v 3195	. . . . . . . . 9 ⊢ (∃𝑔 ∈ 𝐵 (∃ℎ ∈ 𝐶 𝑡 = (𝑔𝐺ℎ) ∧ 𝑥 = (𝑓𝐺𝑡)) ↔ (∃𝑔 ∈ 𝐵 ∃ℎ ∈ 𝐶 𝑡 = (𝑔𝐺ℎ) ∧ 𝑥 = (𝑓𝐺𝑡)))
28	26, 27	bitri 278	. . . . . . . 8 ⊢ (∃𝑔 ∈ 𝐵 ∃ℎ ∈ 𝐶 (𝑡 = (𝑔𝐺ℎ) ∧ 𝑥 = ((𝑓𝐺𝑔)𝐺ℎ)) ↔ (∃𝑔 ∈ 𝐵 ∃ℎ ∈ 𝐶 𝑡 = (𝑔𝐺ℎ) ∧ 𝑥 = (𝑓𝐺𝑡)))
29	28	exbii 1871	. . . . . . 7 ⊢ (∃𝑡∃𝑔 ∈ 𝐵 ∃ℎ ∈ 𝐶 (𝑡 = (𝑔𝐺ℎ) ∧ 𝑥 = ((𝑓𝐺𝑔)𝐺ℎ)) ↔ ∃𝑡(∃𝑔 ∈ 𝐵 ∃ℎ ∈ 𝐶 𝑡 = (𝑔𝐺ℎ) ∧ 𝑥 = (𝑓𝐺𝑡)))
30	16, 17, 29	3bitri 300	. . . . . 6 ⊢ (∃𝑔 ∈ 𝐵 ∃ℎ ∈ 𝐶 𝑥 = ((𝑓𝐺𝑔)𝐺ℎ) ↔ ∃𝑡(∃𝑔 ∈ 𝐵 ∃ℎ ∈ 𝐶 𝑡 = (𝑔𝐺ℎ) ∧ 𝑥 = (𝑓𝐺𝑡)))
31	6, 7, 30	3bitr4g 317	. . . . 5 ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P ∧ 𝐶 ∈ P) → (∃𝑡 ∈ (𝐵𝐹𝐶)𝑥 = (𝑓𝐺𝑡) ↔ ∃𝑔 ∈ 𝐵 ∃ℎ ∈ 𝐶 𝑥 = ((𝑓𝐺𝑔)𝐺ℎ)))
32	31	rexbidv 3189	. . . 4 ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P ∧ 𝐶 ∈ P) → (∃𝑓 ∈ 𝐴 ∃𝑡 ∈ (𝐵𝐹𝐶)𝑥 = (𝑓𝐺𝑡) ↔ ∃𝑓 ∈ 𝐴 ∃𝑔 ∈ 𝐵 ∃ℎ ∈ 𝐶 𝑥 = ((𝑓𝐺𝑔)𝐺ℎ)))
33		genpass.5	. . . . . . 7 ⊢ ((𝑓 ∈ P ∧ 𝑔 ∈ P) → (𝑓𝐹𝑔) ∈ P)
34	33	caovcl 7594	. . . . . 6 ⊢ ((𝐵 ∈ P ∧ 𝐶 ∈ P) → (𝐵𝐹𝐶) ∈ P)
35	1, 2	genpelv 10973	. . . . . 6 ⊢ ((𝐴 ∈ P ∧ (𝐵𝐹𝐶) ∈ P) → (𝑥 ∈ (𝐴𝐹(𝐵𝐹𝐶)) ↔ ∃𝑓 ∈ 𝐴 ∃𝑡 ∈ (𝐵𝐹𝐶)𝑥 = (𝑓𝐺𝑡)))
36	34, 35	sylan2 604	. . . . 5 ⊢ ((𝐴 ∈ P ∧ (𝐵 ∈ P ∧ 𝐶 ∈ P)) → (𝑥 ∈ (𝐴𝐹(𝐵𝐹𝐶)) ↔ ∃𝑓 ∈ 𝐴 ∃𝑡 ∈ (𝐵𝐹𝐶)𝑥 = (𝑓𝐺𝑡)))
37	36	3impb 1130	. . . 4 ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P ∧ 𝐶 ∈ P) → (𝑥 ∈ (𝐴𝐹(𝐵𝐹𝐶)) ↔ ∃𝑓 ∈ 𝐴 ∃𝑡 ∈ (𝐵𝐹𝐶)𝑥 = (𝑓𝐺𝑡)))
38	33	caovcl 7594	. . . . . 6 ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P) → (𝐴𝐹𝐵) ∈ P)
39	1, 2	genpelv 10973	. . . . . 6 ⊢ (((𝐴𝐹𝐵) ∈ P ∧ 𝐶 ∈ P) → (𝑥 ∈ ((𝐴𝐹𝐵)𝐹𝐶) ↔ ∃𝑡 ∈ (𝐴𝐹𝐵)∃ℎ ∈ 𝐶 𝑥 = (𝑡𝐺ℎ)))
40	38, 39	stoic3 1799	. . . . 5 ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P ∧ 𝐶 ∈ P) → (𝑥 ∈ ((𝐴𝐹𝐵)𝐹𝐶) ↔ ∃𝑡 ∈ (𝐴𝐹𝐵)∃ℎ ∈ 𝐶 𝑥 = (𝑡𝐺ℎ)))
41	1, 2	genpelv 10973	. . . . . . . . 9 ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P) → (𝑡 ∈ (𝐴𝐹𝐵) ↔ ∃𝑓 ∈ 𝐴 ∃𝑔 ∈ 𝐵 𝑡 = (𝑓𝐺𝑔)))
42	41	3adant3 1148	. . . . . . . 8 ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P ∧ 𝐶 ∈ P) → (𝑡 ∈ (𝐴𝐹𝐵) ↔ ∃𝑓 ∈ 𝐴 ∃𝑔 ∈ 𝐵 𝑡 = (𝑓𝐺𝑔)))
43	42	anbi1d 642	. . . . . . 7 ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P ∧ 𝐶 ∈ P) → ((𝑡 ∈ (𝐴𝐹𝐵) ∧ ∃ℎ ∈ 𝐶 𝑥 = (𝑡𝐺ℎ)) ↔ (∃𝑓 ∈ 𝐴 ∃𝑔 ∈ 𝐵 𝑡 = (𝑓𝐺𝑔) ∧ ∃ℎ ∈ 𝐶 𝑥 = (𝑡𝐺ℎ))))
44	43	exbidv 1944	. . . . . 6 ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P ∧ 𝐶 ∈ P) → (∃𝑡(𝑡 ∈ (𝐴𝐹𝐵) ∧ ∃ℎ ∈ 𝐶 𝑥 = (𝑡𝐺ℎ)) ↔ ∃𝑡(∃𝑓 ∈ 𝐴 ∃𝑔 ∈ 𝐵 𝑡 = (𝑓𝐺𝑔) ∧ ∃ℎ ∈ 𝐶 𝑥 = (𝑡𝐺ℎ))))
45		df-rex 3090	. . . . . 6 ⊢ (∃𝑡 ∈ (𝐴𝐹𝐵)∃ℎ ∈ 𝐶 𝑥 = (𝑡𝐺ℎ) ↔ ∃𝑡(𝑡 ∈ (𝐴𝐹𝐵) ∧ ∃ℎ ∈ 𝐶 𝑥 = (𝑡𝐺ℎ)))
46		19.41v 1972	. . . . . . . . . . 11 ⊢ (∃𝑡(𝑡 = (𝑓𝐺𝑔) ∧ ∃ℎ ∈ 𝐶 𝑥 = ((𝑓𝐺𝑔)𝐺ℎ)) ↔ (∃𝑡 𝑡 = (𝑓𝐺𝑔) ∧ ∃ℎ ∈ 𝐶 𝑥 = ((𝑓𝐺𝑔)𝐺ℎ)))
47		oveq1 7407	. . . . . . . . . . . . . . 15 ⊢ (𝑡 = (𝑓𝐺𝑔) → (𝑡𝐺ℎ) = ((𝑓𝐺𝑔)𝐺ℎ))
48	47	eqeq2d 2776	. . . . . . . . . . . . . 14 ⊢ (𝑡 = (𝑓𝐺𝑔) → (𝑥 = (𝑡𝐺ℎ) ↔ 𝑥 = ((𝑓𝐺𝑔)𝐺ℎ)))
49	48	rexbidv 3189	. . . . . . . . . . . . 13 ⊢ (𝑡 = (𝑓𝐺𝑔) → (∃ℎ ∈ 𝐶 𝑥 = (𝑡𝐺ℎ) ↔ ∃ℎ ∈ 𝐶 𝑥 = ((𝑓𝐺𝑔)𝐺ℎ)))
50	49	pm5.32i 584	. . . . . . . . . . . 12 ⊢ ((𝑡 = (𝑓𝐺𝑔) ∧ ∃ℎ ∈ 𝐶 𝑥 = (𝑡𝐺ℎ)) ↔ (𝑡 = (𝑓𝐺𝑔) ∧ ∃ℎ ∈ 𝐶 𝑥 = ((𝑓𝐺𝑔)𝐺ℎ)))
51	50	exbii 1871	. . . . . . . . . . 11 ⊢ (∃𝑡(𝑡 = (𝑓𝐺𝑔) ∧ ∃ℎ ∈ 𝐶 𝑥 = (𝑡𝐺ℎ)) ↔ ∃𝑡(𝑡 = (𝑓𝐺𝑔) ∧ ∃ℎ ∈ 𝐶 𝑥 = ((𝑓𝐺𝑔)𝐺ℎ)))
52		ovex 7433	. . . . . . . . . . . . 13 ⊢ (𝑓𝐺𝑔) ∈ V
53	52	isseti 3475	. . . . . . . . . . . 12 ⊢ ∃𝑡 𝑡 = (𝑓𝐺𝑔)
54	53	biantrur 539	. . . . . . . . . . 11 ⊢ (∃ℎ ∈ 𝐶 𝑥 = ((𝑓𝐺𝑔)𝐺ℎ) ↔ (∃𝑡 𝑡 = (𝑓𝐺𝑔) ∧ ∃ℎ ∈ 𝐶 𝑥 = ((𝑓𝐺𝑔)𝐺ℎ)))
55	46, 51, 54	3bitr4ri 307	. . . . . . . . . 10 ⊢ (∃ℎ ∈ 𝐶 𝑥 = ((𝑓𝐺𝑔)𝐺ℎ) ↔ ∃𝑡(𝑡 = (𝑓𝐺𝑔) ∧ ∃ℎ ∈ 𝐶 𝑥 = (𝑡𝐺ℎ)))
56	55	rexbii 3112	. . . . . . . . 9 ⊢ (∃𝑔 ∈ 𝐵 ∃ℎ ∈ 𝐶 𝑥 = ((𝑓𝐺𝑔)𝐺ℎ) ↔ ∃𝑔 ∈ 𝐵 ∃𝑡(𝑡 = (𝑓𝐺𝑔) ∧ ∃ℎ ∈ 𝐶 𝑥 = (𝑡𝐺ℎ)))
57		rexcom4 3292	. . . . . . . . 9 ⊢ (∃𝑔 ∈ 𝐵 ∃𝑡(𝑡 = (𝑓𝐺𝑔) ∧ ∃ℎ ∈ 𝐶 𝑥 = (𝑡𝐺ℎ)) ↔ ∃𝑡∃𝑔 ∈ 𝐵 (𝑡 = (𝑓𝐺𝑔) ∧ ∃ℎ ∈ 𝐶 𝑥 = (𝑡𝐺ℎ)))
58	56, 57	bitri 278	. . . . . . . 8 ⊢ (∃𝑔 ∈ 𝐵 ∃ℎ ∈ 𝐶 𝑥 = ((𝑓𝐺𝑔)𝐺ℎ) ↔ ∃𝑡∃𝑔 ∈ 𝐵 (𝑡 = (𝑓𝐺𝑔) ∧ ∃ℎ ∈ 𝐶 𝑥 = (𝑡𝐺ℎ)))
59	58	rexbii 3112	. . . . . . 7 ⊢ (∃𝑓 ∈ 𝐴 ∃𝑔 ∈ 𝐵 ∃ℎ ∈ 𝐶 𝑥 = ((𝑓𝐺𝑔)𝐺ℎ) ↔ ∃𝑓 ∈ 𝐴 ∃𝑡∃𝑔 ∈ 𝐵 (𝑡 = (𝑓𝐺𝑔) ∧ ∃ℎ ∈ 𝐶 𝑥 = (𝑡𝐺ℎ)))
60		rexcom4 3292	. . . . . . 7 ⊢ (∃𝑓 ∈ 𝐴 ∃𝑡∃𝑔 ∈ 𝐵 (𝑡 = (𝑓𝐺𝑔) ∧ ∃ℎ ∈ 𝐶 𝑥 = (𝑡𝐺ℎ)) ↔ ∃𝑡∃𝑓 ∈ 𝐴 ∃𝑔 ∈ 𝐵 (𝑡 = (𝑓𝐺𝑔) ∧ ∃ℎ ∈ 𝐶 𝑥 = (𝑡𝐺ℎ)))
61		r19.41vv 3235	. . . . . . . 8 ⊢ (∃𝑓 ∈ 𝐴 ∃𝑔 ∈ 𝐵 (𝑡 = (𝑓𝐺𝑔) ∧ ∃ℎ ∈ 𝐶 𝑥 = (𝑡𝐺ℎ)) ↔ (∃𝑓 ∈ 𝐴 ∃𝑔 ∈ 𝐵 𝑡 = (𝑓𝐺𝑔) ∧ ∃ℎ ∈ 𝐶 𝑥 = (𝑡𝐺ℎ)))
62	61	exbii 1871	. . . . . . 7 ⊢ (∃𝑡∃𝑓 ∈ 𝐴 ∃𝑔 ∈ 𝐵 (𝑡 = (𝑓𝐺𝑔) ∧ ∃ℎ ∈ 𝐶 𝑥 = (𝑡𝐺ℎ)) ↔ ∃𝑡(∃𝑓 ∈ 𝐴 ∃𝑔 ∈ 𝐵 𝑡 = (𝑓𝐺𝑔) ∧ ∃ℎ ∈ 𝐶 𝑥 = (𝑡𝐺ℎ)))
63	59, 60, 62	3bitri 300	. . . . . 6 ⊢ (∃𝑓 ∈ 𝐴 ∃𝑔 ∈ 𝐵 ∃ℎ ∈ 𝐶 𝑥 = ((𝑓𝐺𝑔)𝐺ℎ) ↔ ∃𝑡(∃𝑓 ∈ 𝐴 ∃𝑔 ∈ 𝐵 𝑡 = (𝑓𝐺𝑔) ∧ ∃ℎ ∈ 𝐶 𝑥 = (𝑡𝐺ℎ)))
64	44, 45, 63	3bitr4g 317	. . . . 5 ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P ∧ 𝐶 ∈ P) → (∃𝑡 ∈ (𝐴𝐹𝐵)∃ℎ ∈ 𝐶 𝑥 = (𝑡𝐺ℎ) ↔ ∃𝑓 ∈ 𝐴 ∃𝑔 ∈ 𝐵 ∃ℎ ∈ 𝐶 𝑥 = ((𝑓𝐺𝑔)𝐺ℎ)))
65	40, 64	bitrd 282	. . . 4 ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P ∧ 𝐶 ∈ P) → (𝑥 ∈ ((𝐴𝐹𝐵)𝐹𝐶) ↔ ∃𝑓 ∈ 𝐴 ∃𝑔 ∈ 𝐵 ∃ℎ ∈ 𝐶 𝑥 = ((𝑓𝐺𝑔)𝐺ℎ)))
66	32, 37, 65	3bitr4rd 315	. . 3 ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P ∧ 𝐶 ∈ P) → (𝑥 ∈ ((𝐴𝐹𝐵)𝐹𝐶) ↔ 𝑥 ∈ (𝐴𝐹(𝐵𝐹𝐶))))
67	66	eqrdv 2763	. 2 ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P ∧ 𝐶 ∈ P) → ((𝐴𝐹𝐵)𝐹𝐶) = (𝐴𝐹(𝐵𝐹𝐶)))
68		genpass.4	. . 3 ⊢ dom 𝐹 = (P × P)
69		0npr 10965	. . 3 ⊢ ¬ ∅ ∈ P
70	68, 69	ndmovass 7588	. 2 ⊢ (¬ (𝐴 ∈ P ∧ 𝐵 ∈ P ∧ 𝐶 ∈ P) → ((𝐴𝐹𝐵)𝐹𝐶) = (𝐴𝐹(𝐵𝐹𝐶)))
71	67, 70	pm2.61i 184	1 ⊢ ((𝐴𝐹𝐵)𝐹𝐶) = (𝐴𝐹(𝐵𝐹𝐶))

Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 400   ∧ w3a 1101   = wceq 1563  ∃wex 1802   ∈ wcel 2145  {cab 2743  ∃wrex 3089   × cxp 5650  dom cdm 5652  (class class class)co 7400   ∈ cmpo 7402  Qcnq 10825  Pcnp 10832

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1818  ax-4 1832  ax-5 1933  ax-6 1990  ax-7 2031  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2215  ax-ext 2737  ax-sep 5251  ax-nul 5261  ax-pow 5327  ax-pr 5395  ax-un 7722  ax-inf2 9598

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3or 1102  df-3an 1103  df-tru 1566  df-fal 1576  df-ex 1803  df-nf 1807  df-sb 2094  df-mo 2569  df-eu 2599  df-clab 2744  df-cleq 2757  df-clel 2840  df-nfc 2914  df-ne 2961  df-ral 3080  df-rex 3090  df-rab 3418  df-v 3459  df-sbc 3748  df-dif 3910  df-un 3912  df-in 3914  df-ss 3924  df-pss 3927  df-nul 4289  df-if 4484  df-pw 4560  df-sn 4586  df-pr 4588  df-op 4592  df-uni 4869  df-br 5106  df-opab 5168  df-tr 5213  df-id 5547  df-eprel 5552  df-po 5560  df-so 5561  df-fr 5605  df-we 5607  df-xp 5658  df-rel 5659  df-cnv 5660  df-co 5661  df-dm 5662  df-ord 6353  df-on 6354  df-lim 6355  df-suc 6356  df-iota 6481  df-fun 6527  df-fv 6533  df-ov 7403  df-oprab 7404  df-mpo 7405  df-om 7851  df-ni 10845  df-nq 10885  df-np 10954

This theorem is referenced by:  addasspr  10995  mulasspr  10997