Theorem genpass 11066

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 11057	. . . . . . . . 9 ⊢ ((𝐵 ∈ P ∧ 𝐶 ∈ P) → (𝑡 ∈ (𝐵𝐹𝐶) ↔ ∃𝑔 ∈ 𝐵 ∃ℎ ∈ 𝐶 𝑡 = (𝑔𝐺ℎ)))
4	3	3adant1 1148	. . . . . . . 8 ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P ∧ 𝐶 ∈ P) → (𝑡 ∈ (𝐵𝐹𝐶) ↔ ∃𝑔 ∈ 𝐵 ∃ℎ ∈ 𝐶 𝑡 = (𝑔𝐺ℎ)))
5	4	anbi1d 643	. . . . . . 7 ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P ∧ 𝐶 ∈ P) → ((𝑡 ∈ (𝐵𝐹𝐶) ∧ 𝑥 = (𝑓𝐺𝑡)) ↔ (∃𝑔 ∈ 𝐵 ∃ℎ ∈ 𝐶 𝑡 = (𝑔𝐺ℎ) ∧ 𝑥 = (𝑓𝐺𝑡))))
6	5	exbidv 1954	. . . . . 6 ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P ∧ 𝐶 ∈ P) → (∃𝑡(𝑡 ∈ (𝐵𝐹𝐶) ∧ 𝑥 = (𝑓𝐺𝑡)) ↔ ∃𝑡(∃𝑔 ∈ 𝐵 ∃ℎ ∈ 𝐶 𝑡 = (𝑔𝐺ℎ) ∧ 𝑥 = (𝑓𝐺𝑡))))
7		df-rex 3087	. . . . . 6 ⊢ (∃𝑡 ∈ (𝐵𝐹𝐶)𝑥 = (𝑓𝐺𝑡) ↔ ∃𝑡(𝑡 ∈ (𝐵𝐹𝐶) ∧ 𝑥 = (𝑓𝐺𝑡)))
8		ovex 7441	. . . . . . . . . . . . 13 ⊢ (𝑔𝐺ℎ) ∈ V
9	8	isseti 3468	. . . . . . . . . . . 12 ⊢ ∃𝑡 𝑡 = (𝑔𝐺ℎ)
10	9	biantrur 540	. . . . . . . . . . 11 ⊢ (𝑥 = ((𝑓𝐺𝑔)𝐺ℎ) ↔ (∃𝑡 𝑡 = (𝑔𝐺ℎ) ∧ 𝑥 = ((𝑓𝐺𝑔)𝐺ℎ)))
11		19.41v 1982	. . . . . . . . . . 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 3109	. . . . . . . . 9 ⊢ (∃ℎ ∈ 𝐶 𝑥 = ((𝑓𝐺𝑔)𝐺ℎ) ↔ ∃ℎ ∈ 𝐶 ∃𝑡(𝑡 = (𝑔𝐺ℎ) ∧ 𝑥 = ((𝑓𝐺𝑔)𝐺ℎ)))
14		rexcom4 3289	. . . . . . . . 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 3109	. . . . . . 7 ⊢ (∃𝑔 ∈ 𝐵 ∃ℎ ∈ 𝐶 𝑥 = ((𝑓𝐺𝑔)𝐺ℎ) ↔ ∃𝑔 ∈ 𝐵 ∃𝑡∃ℎ ∈ 𝐶 (𝑡 = (𝑔𝐺ℎ) ∧ 𝑥 = ((𝑓𝐺𝑔)𝐺ℎ)))
17		rexcom4 3289	. . . . . . 7 ⊢ (∃𝑔 ∈ 𝐵 ∃𝑡∃ℎ ∈ 𝐶 (𝑡 = (𝑔𝐺ℎ) ∧ 𝑥 = ((𝑓𝐺𝑔)𝐺ℎ)) ↔ ∃𝑡∃𝑔 ∈ 𝐵 ∃ℎ ∈ 𝐶 (𝑡 = (𝑔𝐺ℎ) ∧ 𝑥 = ((𝑓𝐺𝑔)𝐺ℎ)))
18		oveq2 7416	. . . . . . . . . . . . . . 15 ⊢ (𝑡 = (𝑔𝐺ℎ) → (𝑓𝐺𝑡) = (𝑓𝐺(𝑔𝐺ℎ)))
19		genpass.6	. . . . . . . . . . . . . . 15 ⊢ ((𝑓𝐺𝑔)𝐺ℎ) = (𝑓𝐺(𝑔𝐺ℎ))
20	18, 19	eqtr4di 2813	. . . . . . . . . . . . . 14 ⊢ (𝑡 = (𝑔𝐺ℎ) → (𝑓𝐺𝑡) = ((𝑓𝐺𝑔)𝐺ℎ))
21	20	eqeq2d 2771	. . . . . . . . . . . . 13 ⊢ (𝑡 = (𝑔𝐺ℎ) → (𝑥 = (𝑓𝐺𝑡) ↔ 𝑥 = ((𝑓𝐺𝑔)𝐺ℎ)))
22	21	pm5.32i 585	. . . . . . . . . . . 12 ⊢ ((𝑡 = (𝑔𝐺ℎ) ∧ 𝑥 = (𝑓𝐺𝑡)) ↔ (𝑡 = (𝑔𝐺ℎ) ∧ 𝑥 = ((𝑓𝐺𝑔)𝐺ℎ)))
23	22	rexbii 3109	. . . . . . . . . . 11 ⊢ (∃ℎ ∈ 𝐶 (𝑡 = (𝑔𝐺ℎ) ∧ 𝑥 = (𝑓𝐺𝑡)) ↔ ∃ℎ ∈ 𝐶 (𝑡 = (𝑔𝐺ℎ) ∧ 𝑥 = ((𝑓𝐺𝑔)𝐺ℎ)))
24		r19.41v 3192	. . . . . . . . . . 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 3109	. . . . . . . . 9 ⊢ (∃𝑔 ∈ 𝐵 ∃ℎ ∈ 𝐶 (𝑡 = (𝑔𝐺ℎ) ∧ 𝑥 = ((𝑓𝐺𝑔)𝐺ℎ)) ↔ ∃𝑔 ∈ 𝐵 (∃ℎ ∈ 𝐶 𝑡 = (𝑔𝐺ℎ) ∧ 𝑥 = (𝑓𝐺𝑡)))
27		r19.41v 3192	. . . . . . . . 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 1881	. . . . . . 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 3186	. . . 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 7603	. . . . . 6 ⊢ ((𝐵 ∈ P ∧ 𝐶 ∈ P) → (𝐵𝐹𝐶) ∈ P)
35	1, 2	genpelv 11057	. . . . . 6 ⊢ ((𝐴 ∈ P ∧ (𝐵𝐹𝐶) ∈ P) → (𝑥 ∈ (𝐴𝐹(𝐵𝐹𝐶)) ↔ ∃𝑓 ∈ 𝐴 ∃𝑡 ∈ (𝐵𝐹𝐶)𝑥 = (𝑓𝐺𝑡)))
36	34, 35	sylan2 605	. . . . 5 ⊢ ((𝐴 ∈ P ∧ (𝐵 ∈ P ∧ 𝐶 ∈ P)) → (𝑥 ∈ (𝐴𝐹(𝐵𝐹𝐶)) ↔ ∃𝑓 ∈ 𝐴 ∃𝑡 ∈ (𝐵𝐹𝐶)𝑥 = (𝑓𝐺𝑡)))
37	36	3impb 1132	. . . 4 ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P ∧ 𝐶 ∈ P) → (𝑥 ∈ (𝐴𝐹(𝐵𝐹𝐶)) ↔ ∃𝑓 ∈ 𝐴 ∃𝑡 ∈ (𝐵𝐹𝐶)𝑥 = (𝑓𝐺𝑡)))
38	33	caovcl 7603	. . . . . 6 ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P) → (𝐴𝐹𝐵) ∈ P)
39	1, 2	genpelv 11057	. . . . . 6 ⊢ (((𝐴𝐹𝐵) ∈ P ∧ 𝐶 ∈ P) → (𝑥 ∈ ((𝐴𝐹𝐵)𝐹𝐶) ↔ ∃𝑡 ∈ (𝐴𝐹𝐵)∃ℎ ∈ 𝐶 𝑥 = (𝑡𝐺ℎ)))
40	38, 39	stoic3 1809	. . . . 5 ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P ∧ 𝐶 ∈ P) → (𝑥 ∈ ((𝐴𝐹𝐵)𝐹𝐶) ↔ ∃𝑡 ∈ (𝐴𝐹𝐵)∃ℎ ∈ 𝐶 𝑥 = (𝑡𝐺ℎ)))
41	1, 2	genpelv 11057	. . . . . . . . 9 ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P) → (𝑡 ∈ (𝐴𝐹𝐵) ↔ ∃𝑓 ∈ 𝐴 ∃𝑔 ∈ 𝐵 𝑡 = (𝑓𝐺𝑔)))
42	41	3adant3 1150	. . . . . . . 8 ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P ∧ 𝐶 ∈ P) → (𝑡 ∈ (𝐴𝐹𝐵) ↔ ∃𝑓 ∈ 𝐴 ∃𝑔 ∈ 𝐵 𝑡 = (𝑓𝐺𝑔)))
43	42	anbi1d 643	. . . . . . 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 1954	. . . . . 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 3087	. . . . . 6 ⊢ (∃𝑡 ∈ (𝐴𝐹𝐵)∃ℎ ∈ 𝐶 𝑥 = (𝑡𝐺ℎ) ↔ ∃𝑡(𝑡 ∈ (𝐴𝐹𝐵) ∧ ∃ℎ ∈ 𝐶 𝑥 = (𝑡𝐺ℎ)))
46		19.41v 1982	. . . . . . . . . . 11 ⊢ (∃𝑡(𝑡 = (𝑓𝐺𝑔) ∧ ∃ℎ ∈ 𝐶 𝑥 = ((𝑓𝐺𝑔)𝐺ℎ)) ↔ (∃𝑡 𝑡 = (𝑓𝐺𝑔) ∧ ∃ℎ ∈ 𝐶 𝑥 = ((𝑓𝐺𝑔)𝐺ℎ)))
47		oveq1 7415	. . . . . . . . . . . . . . 15 ⊢ (𝑡 = (𝑓𝐺𝑔) → (𝑡𝐺ℎ) = ((𝑓𝐺𝑔)𝐺ℎ))
48	47	eqeq2d 2771	. . . . . . . . . . . . . 14 ⊢ (𝑡 = (𝑓𝐺𝑔) → (𝑥 = (𝑡𝐺ℎ) ↔ 𝑥 = ((𝑓𝐺𝑔)𝐺ℎ)))
49	48	rexbidv 3186	. . . . . . . . . . . . 13 ⊢ (𝑡 = (𝑓𝐺𝑔) → (∃ℎ ∈ 𝐶 𝑥 = (𝑡𝐺ℎ) ↔ ∃ℎ ∈ 𝐶 𝑥 = ((𝑓𝐺𝑔)𝐺ℎ)))
50	49	pm5.32i 585	. . . . . . . . . . . 12 ⊢ ((𝑡 = (𝑓𝐺𝑔) ∧ ∃ℎ ∈ 𝐶 𝑥 = (𝑡𝐺ℎ)) ↔ (𝑡 = (𝑓𝐺𝑔) ∧ ∃ℎ ∈ 𝐶 𝑥 = ((𝑓𝐺𝑔)𝐺ℎ)))
51	50	exbii 1881	. . . . . . . . . . 11 ⊢ (∃𝑡(𝑡 = (𝑓𝐺𝑔) ∧ ∃ℎ ∈ 𝐶 𝑥 = (𝑡𝐺ℎ)) ↔ ∃𝑡(𝑡 = (𝑓𝐺𝑔) ∧ ∃ℎ ∈ 𝐶 𝑥 = ((𝑓𝐺𝑔)𝐺ℎ)))
52		ovex 7441	. . . . . . . . . . . . 13 ⊢ (𝑓𝐺𝑔) ∈ V
53	52	isseti 3468	. . . . . . . . . . . 12 ⊢ ∃𝑡 𝑡 = (𝑓𝐺𝑔)
54	53	biantrur 540	. . . . . . . . . . 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 3109	. . . . . . . . 9 ⊢ (∃𝑔 ∈ 𝐵 ∃ℎ ∈ 𝐶 𝑥 = ((𝑓𝐺𝑔)𝐺ℎ) ↔ ∃𝑔 ∈ 𝐵 ∃𝑡(𝑡 = (𝑓𝐺𝑔) ∧ ∃ℎ ∈ 𝐶 𝑥 = (𝑡𝐺ℎ)))
57		rexcom4 3289	. . . . . . . . 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 3109	. . . . . . 7 ⊢ (∃𝑓 ∈ 𝐴 ∃𝑔 ∈ 𝐵 ∃ℎ ∈ 𝐶 𝑥 = ((𝑓𝐺𝑔)𝐺ℎ) ↔ ∃𝑓 ∈ 𝐴 ∃𝑡∃𝑔 ∈ 𝐵 (𝑡 = (𝑓𝐺𝑔) ∧ ∃ℎ ∈ 𝐶 𝑥 = (𝑡𝐺ℎ)))
60		rexcom4 3289	. . . . . . 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 3232	. . . . . . . 8 ⊢ (∃𝑓 ∈ 𝐴 ∃𝑔 ∈ 𝐵 (𝑡 = (𝑓𝐺𝑔) ∧ ∃ℎ ∈ 𝐶 𝑥 = (𝑡𝐺ℎ)) ↔ (∃𝑓 ∈ 𝐴 ∃𝑔 ∈ 𝐵 𝑡 = (𝑓𝐺𝑔) ∧ ∃ℎ ∈ 𝐶 𝑥 = (𝑡𝐺ℎ)))
62	61	exbii 1881	. . . . . . 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 2758	. 2 ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P ∧ 𝐶 ∈ P) → ((𝐴𝐹𝐵)𝐹𝐶) = (𝐴𝐹(𝐵𝐹𝐶)))
68		genpass.4	. . 3 ⊢ dom 𝐹 = (P × P)
69		0npr 11049	. . 3 ⊢ ¬ ∅ ∈ P
70	68, 69	ndmovass 7597	. 2 ⊢ (¬ (𝐴 ∈ P ∧ 𝐵 ∈ P ∧ 𝐶 ∈ P) → ((𝐴𝐹𝐵)𝐹𝐶) = (𝐴𝐹(𝐵𝐹𝐶)))
71	67, 70	pm2.61i 184	1 ⊢ ((𝐴𝐹𝐵)𝐹𝐶) = (𝐴𝐹(𝐵𝐹𝐶))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   ∧ w3a 1103   = wceq 1570  ∃wex 1812   ∈ wcel 2145  {cab 2738  ∃wrex 3086   × cxp 5645  dom cdm 5647  (class class class)co 7408   ∈ cmpo 7410  Qcnq 10909  Pcnp 10916

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-10 2178  ax-11 2194  ax-12 2213  ax-ext 2732  ax-sep 5248  ax-nul 5259  ax-pow 5326  ax-pr 5390  ax-un 7734  ax-inf2 9620

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-ral 3077  df-rex 3087  df-rab 3413  df-v 3452  df-sbc 3739  df-dif 3901  df-un 3903  df-in 3905  df-ss 3915  df-pss 3918  df-nul 4279  df-if 4482  df-pw 4558  df-sn 4584  df-pr 4586  df-op 4590  df-uni 4867  df-br 5103  df-opab 5167  df-tr 5212  df-id 5542  df-eprel 5547  df-po 5555  df-so 5556  df-fr 5600  df-we 5602  df-xp 5653  df-rel 5654  df-cnv 5655  df-co 5656  df-dm 5657  df-ord 6354  df-on 6355  df-lim 6356  df-suc 6357  df-iota 6483  df-fun 6529  df-fv 6535  df-ov 7411  df-oprab 7412  df-mpo 7413  df-om 7861  df-ni 10929  df-nq 10969  df-np 11038

This theorem is used by:  addasspr  11079  mulasspr  11081