Theorem genpass 10957

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 10948	. . . . . . . . 9 ⊢ ((𝐵 ∈ P ∧ 𝐶 ∈ P) → (𝑡 ∈ (𝐵𝐹𝐶) ↔ ∃𝑔 ∈ 𝐵 ∃ℎ ∈ 𝐶 𝑡 = (𝑔𝐺ℎ)))
4	3	3adant1 1139	. . . . . . . 8 ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P ∧ 𝐶 ∈ P) → (𝑡 ∈ (𝐵𝐹𝐶) ↔ ∃𝑔 ∈ 𝐵 ∃ℎ ∈ 𝐶 𝑡 = (𝑔𝐺ℎ)))
5	4	anbi1d 639	. . . . . . 7 ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P ∧ 𝐶 ∈ P) → ((𝑡 ∈ (𝐵𝐹𝐶) ∧ 𝑥 = (𝑓𝐺𝑡)) ↔ (∃𝑔 ∈ 𝐵 ∃ℎ ∈ 𝐶 𝑡 = (𝑔𝐺ℎ) ∧ 𝑥 = (𝑓𝐺𝑡))))
6	5	exbidv 1935	. . . . . 6 ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P ∧ 𝐶 ∈ P) → (∃𝑡(𝑡 ∈ (𝐵𝐹𝐶) ∧ 𝑥 = (𝑓𝐺𝑡)) ↔ ∃𝑡(∃𝑔 ∈ 𝐵 ∃ℎ ∈ 𝐶 𝑡 = (𝑔𝐺ℎ) ∧ 𝑥 = (𝑓𝐺𝑡))))
7		df-rex 3081	. . . . . 6 ⊢ (∃𝑡 ∈ (𝐵𝐹𝐶)𝑥 = (𝑓𝐺𝑡) ↔ ∃𝑡(𝑡 ∈ (𝐵𝐹𝐶) ∧ 𝑥 = (𝑓𝐺𝑡)))
8		ovex 7418	. . . . . . . . . . . . 13 ⊢ (𝑔𝐺ℎ) ∈ V
9	8	isseti 3466	. . . . . . . . . . . 12 ⊢ ∃𝑡 𝑡 = (𝑔𝐺ℎ)
10	9	biantrur 537	. . . . . . . . . . 11 ⊢ (𝑥 = ((𝑓𝐺𝑔)𝐺ℎ) ↔ (∃𝑡 𝑡 = (𝑔𝐺ℎ) ∧ 𝑥 = ((𝑓𝐺𝑔)𝐺ℎ)))
11		19.41v 1963	. . . . . . . . . . 11 ⊢ (∃𝑡(𝑡 = (𝑔𝐺ℎ) ∧ 𝑥 = ((𝑓𝐺𝑔)𝐺ℎ)) ↔ (∃𝑡 𝑡 = (𝑔𝐺ℎ) ∧ 𝑥 = ((𝑓𝐺𝑔)𝐺ℎ)))
12	10, 11	bitr4i 280	. . . . . . . . . 10 ⊢ (𝑥 = ((𝑓𝐺𝑔)𝐺ℎ) ↔ ∃𝑡(𝑡 = (𝑔𝐺ℎ) ∧ 𝑥 = ((𝑓𝐺𝑔)𝐺ℎ)))
13	12	rexbii 3103	. . . . . . . . 9 ⊢ (∃ℎ ∈ 𝐶 𝑥 = ((𝑓𝐺𝑔)𝐺ℎ) ↔ ∃ℎ ∈ 𝐶 ∃𝑡(𝑡 = (𝑔𝐺ℎ) ∧ 𝑥 = ((𝑓𝐺𝑔)𝐺ℎ)))
14		rexcom4 3283	. . . . . . . . 9 ⊢ (∃ℎ ∈ 𝐶 ∃𝑡(𝑡 = (𝑔𝐺ℎ) ∧ 𝑥 = ((𝑓𝐺𝑔)𝐺ℎ)) ↔ ∃𝑡∃ℎ ∈ 𝐶 (𝑡 = (𝑔𝐺ℎ) ∧ 𝑥 = ((𝑓𝐺𝑔)𝐺ℎ)))
15	13, 14	bitri 277	. . . . . . . 8 ⊢ (∃ℎ ∈ 𝐶 𝑥 = ((𝑓𝐺𝑔)𝐺ℎ) ↔ ∃𝑡∃ℎ ∈ 𝐶 (𝑡 = (𝑔𝐺ℎ) ∧ 𝑥 = ((𝑓𝐺𝑔)𝐺ℎ)))
16	15	rexbii 3103	. . . . . . 7 ⊢ (∃𝑔 ∈ 𝐵 ∃ℎ ∈ 𝐶 𝑥 = ((𝑓𝐺𝑔)𝐺ℎ) ↔ ∃𝑔 ∈ 𝐵 ∃𝑡∃ℎ ∈ 𝐶 (𝑡 = (𝑔𝐺ℎ) ∧ 𝑥 = ((𝑓𝐺𝑔)𝐺ℎ)))
17		rexcom4 3283	. . . . . . 7 ⊢ (∃𝑔 ∈ 𝐵 ∃𝑡∃ℎ ∈ 𝐶 (𝑡 = (𝑔𝐺ℎ) ∧ 𝑥 = ((𝑓𝐺𝑔)𝐺ℎ)) ↔ ∃𝑡∃𝑔 ∈ 𝐵 ∃ℎ ∈ 𝐶 (𝑡 = (𝑔𝐺ℎ) ∧ 𝑥 = ((𝑓𝐺𝑔)𝐺ℎ)))
18		oveq2 7393	. . . . . . . . . . . . . . 15 ⊢ (𝑡 = (𝑔𝐺ℎ) → (𝑓𝐺𝑡) = (𝑓𝐺(𝑔𝐺ℎ)))
19		genpass.6	. . . . . . . . . . . . . . 15 ⊢ ((𝑓𝐺𝑔)𝐺ℎ) = (𝑓𝐺(𝑔𝐺ℎ))
20	18, 19	eqtr4di 2809	. . . . . . . . . . . . . 14 ⊢ (𝑡 = (𝑔𝐺ℎ) → (𝑓𝐺𝑡) = ((𝑓𝐺𝑔)𝐺ℎ))
21	20	eqeq2d 2767	. . . . . . . . . . . . 13 ⊢ (𝑡 = (𝑔𝐺ℎ) → (𝑥 = (𝑓𝐺𝑡) ↔ 𝑥 = ((𝑓𝐺𝑔)𝐺ℎ)))
22	21	pm5.32i 581	. . . . . . . . . . . 12 ⊢ ((𝑡 = (𝑔𝐺ℎ) ∧ 𝑥 = (𝑓𝐺𝑡)) ↔ (𝑡 = (𝑔𝐺ℎ) ∧ 𝑥 = ((𝑓𝐺𝑔)𝐺ℎ)))
23	22	rexbii 3103	. . . . . . . . . . 11 ⊢ (∃ℎ ∈ 𝐶 (𝑡 = (𝑔𝐺ℎ) ∧ 𝑥 = (𝑓𝐺𝑡)) ↔ ∃ℎ ∈ 𝐶 (𝑡 = (𝑔𝐺ℎ) ∧ 𝑥 = ((𝑓𝐺𝑔)𝐺ℎ)))
24		r19.41v 3186	. . . . . . . . . . 11 ⊢ (∃ℎ ∈ 𝐶 (𝑡 = (𝑔𝐺ℎ) ∧ 𝑥 = (𝑓𝐺𝑡)) ↔ (∃ℎ ∈ 𝐶 𝑡 = (𝑔𝐺ℎ) ∧ 𝑥 = (𝑓𝐺𝑡)))
25	23, 24	bitr3i 279	. . . . . . . . . 10 ⊢ (∃ℎ ∈ 𝐶 (𝑡 = (𝑔𝐺ℎ) ∧ 𝑥 = ((𝑓𝐺𝑔)𝐺ℎ)) ↔ (∃ℎ ∈ 𝐶 𝑡 = (𝑔𝐺ℎ) ∧ 𝑥 = (𝑓𝐺𝑡)))
26	25	rexbii 3103	. . . . . . . . 9 ⊢ (∃𝑔 ∈ 𝐵 ∃ℎ ∈ 𝐶 (𝑡 = (𝑔𝐺ℎ) ∧ 𝑥 = ((𝑓𝐺𝑔)𝐺ℎ)) ↔ ∃𝑔 ∈ 𝐵 (∃ℎ ∈ 𝐶 𝑡 = (𝑔𝐺ℎ) ∧ 𝑥 = (𝑓𝐺𝑡)))
27		r19.41v 3186	. . . . . . . . 9 ⊢ (∃𝑔 ∈ 𝐵 (∃ℎ ∈ 𝐶 𝑡 = (𝑔𝐺ℎ) ∧ 𝑥 = (𝑓𝐺𝑡)) ↔ (∃𝑔 ∈ 𝐵 ∃ℎ ∈ 𝐶 𝑡 = (𝑔𝐺ℎ) ∧ 𝑥 = (𝑓𝐺𝑡)))
28	26, 27	bitri 277	. . . . . . . 8 ⊢ (∃𝑔 ∈ 𝐵 ∃ℎ ∈ 𝐶 (𝑡 = (𝑔𝐺ℎ) ∧ 𝑥 = ((𝑓𝐺𝑔)𝐺ℎ)) ↔ (∃𝑔 ∈ 𝐵 ∃ℎ ∈ 𝐶 𝑡 = (𝑔𝐺ℎ) ∧ 𝑥 = (𝑓𝐺𝑡)))
29	28	exbii 1862	. . . . . . 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 299	. . . . . 6 ⊢ (∃𝑔 ∈ 𝐵 ∃ℎ ∈ 𝐶 𝑥 = ((𝑓𝐺𝑔)𝐺ℎ) ↔ ∃𝑡(∃𝑔 ∈ 𝐵 ∃ℎ ∈ 𝐶 𝑡 = (𝑔𝐺ℎ) ∧ 𝑥 = (𝑓𝐺𝑡)))
31	6, 7, 30	3bitr4g 316	. . . . 5 ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P ∧ 𝐶 ∈ P) → (∃𝑡 ∈ (𝐵𝐹𝐶)𝑥 = (𝑓𝐺𝑡) ↔ ∃𝑔 ∈ 𝐵 ∃ℎ ∈ 𝐶 𝑥 = ((𝑓𝐺𝑔)𝐺ℎ)))
32	31	rexbidv 3180	. . . 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 7579	. . . . . 6 ⊢ ((𝐵 ∈ P ∧ 𝐶 ∈ P) → (𝐵𝐹𝐶) ∈ P)
35	1, 2	genpelv 10948	. . . . . 6 ⊢ ((𝐴 ∈ P ∧ (𝐵𝐹𝐶) ∈ P) → (𝑥 ∈ (𝐴𝐹(𝐵𝐹𝐶)) ↔ ∃𝑓 ∈ 𝐴 ∃𝑡 ∈ (𝐵𝐹𝐶)𝑥 = (𝑓𝐺𝑡)))
36	34, 35	sylan2 601	. . . . 5 ⊢ ((𝐴 ∈ P ∧ (𝐵 ∈ P ∧ 𝐶 ∈ P)) → (𝑥 ∈ (𝐴𝐹(𝐵𝐹𝐶)) ↔ ∃𝑓 ∈ 𝐴 ∃𝑡 ∈ (𝐵𝐹𝐶)𝑥 = (𝑓𝐺𝑡)))
37	36	3impb 1123	. . . 4 ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P ∧ 𝐶 ∈ P) → (𝑥 ∈ (𝐴𝐹(𝐵𝐹𝐶)) ↔ ∃𝑓 ∈ 𝐴 ∃𝑡 ∈ (𝐵𝐹𝐶)𝑥 = (𝑓𝐺𝑡)))
38	33	caovcl 7579	. . . . . 6 ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P) → (𝐴𝐹𝐵) ∈ P)
39	1, 2	genpelv 10948	. . . . . 6 ⊢ (((𝐴𝐹𝐵) ∈ P ∧ 𝐶 ∈ P) → (𝑥 ∈ ((𝐴𝐹𝐵)𝐹𝐶) ↔ ∃𝑡 ∈ (𝐴𝐹𝐵)∃ℎ ∈ 𝐶 𝑥 = (𝑡𝐺ℎ)))
40	38, 39	stoic3 1790	. . . . 5 ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P ∧ 𝐶 ∈ P) → (𝑥 ∈ ((𝐴𝐹𝐵)𝐹𝐶) ↔ ∃𝑡 ∈ (𝐴𝐹𝐵)∃ℎ ∈ 𝐶 𝑥 = (𝑡𝐺ℎ)))
41	1, 2	genpelv 10948	. . . . . . . . 9 ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P) → (𝑡 ∈ (𝐴𝐹𝐵) ↔ ∃𝑓 ∈ 𝐴 ∃𝑔 ∈ 𝐵 𝑡 = (𝑓𝐺𝑔)))
42	41	3adant3 1141	. . . . . . . 8 ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P ∧ 𝐶 ∈ P) → (𝑡 ∈ (𝐴𝐹𝐵) ↔ ∃𝑓 ∈ 𝐴 ∃𝑔 ∈ 𝐵 𝑡 = (𝑓𝐺𝑔)))
43	42	anbi1d 639	. . . . . . 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 1935	. . . . . 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 3081	. . . . . 6 ⊢ (∃𝑡 ∈ (𝐴𝐹𝐵)∃ℎ ∈ 𝐶 𝑥 = (𝑡𝐺ℎ) ↔ ∃𝑡(𝑡 ∈ (𝐴𝐹𝐵) ∧ ∃ℎ ∈ 𝐶 𝑥 = (𝑡𝐺ℎ)))
46		19.41v 1963	. . . . . . . . . . 11 ⊢ (∃𝑡(𝑡 = (𝑓𝐺𝑔) ∧ ∃ℎ ∈ 𝐶 𝑥 = ((𝑓𝐺𝑔)𝐺ℎ)) ↔ (∃𝑡 𝑡 = (𝑓𝐺𝑔) ∧ ∃ℎ ∈ 𝐶 𝑥 = ((𝑓𝐺𝑔)𝐺ℎ)))
47		oveq1 7392	. . . . . . . . . . . . . . 15 ⊢ (𝑡 = (𝑓𝐺𝑔) → (𝑡𝐺ℎ) = ((𝑓𝐺𝑔)𝐺ℎ))
48	47	eqeq2d 2767	. . . . . . . . . . . . . 14 ⊢ (𝑡 = (𝑓𝐺𝑔) → (𝑥 = (𝑡𝐺ℎ) ↔ 𝑥 = ((𝑓𝐺𝑔)𝐺ℎ)))
49	48	rexbidv 3180	. . . . . . . . . . . . 13 ⊢ (𝑡 = (𝑓𝐺𝑔) → (∃ℎ ∈ 𝐶 𝑥 = (𝑡𝐺ℎ) ↔ ∃ℎ ∈ 𝐶 𝑥 = ((𝑓𝐺𝑔)𝐺ℎ)))
50	49	pm5.32i 581	. . . . . . . . . . . 12 ⊢ ((𝑡 = (𝑓𝐺𝑔) ∧ ∃ℎ ∈ 𝐶 𝑥 = (𝑡𝐺ℎ)) ↔ (𝑡 = (𝑓𝐺𝑔) ∧ ∃ℎ ∈ 𝐶 𝑥 = ((𝑓𝐺𝑔)𝐺ℎ)))
51	50	exbii 1862	. . . . . . . . . . 11 ⊢ (∃𝑡(𝑡 = (𝑓𝐺𝑔) ∧ ∃ℎ ∈ 𝐶 𝑥 = (𝑡𝐺ℎ)) ↔ ∃𝑡(𝑡 = (𝑓𝐺𝑔) ∧ ∃ℎ ∈ 𝐶 𝑥 = ((𝑓𝐺𝑔)𝐺ℎ)))
52		ovex 7418	. . . . . . . . . . . . 13 ⊢ (𝑓𝐺𝑔) ∈ V
53	52	isseti 3466	. . . . . . . . . . . 12 ⊢ ∃𝑡 𝑡 = (𝑓𝐺𝑔)
54	53	biantrur 537	. . . . . . . . . . 11 ⊢ (∃ℎ ∈ 𝐶 𝑥 = ((𝑓𝐺𝑔)𝐺ℎ) ↔ (∃𝑡 𝑡 = (𝑓𝐺𝑔) ∧ ∃ℎ ∈ 𝐶 𝑥 = ((𝑓𝐺𝑔)𝐺ℎ)))
55	46, 51, 54	3bitr4ri 306	. . . . . . . . . 10 ⊢ (∃ℎ ∈ 𝐶 𝑥 = ((𝑓𝐺𝑔)𝐺ℎ) ↔ ∃𝑡(𝑡 = (𝑓𝐺𝑔) ∧ ∃ℎ ∈ 𝐶 𝑥 = (𝑡𝐺ℎ)))
56	55	rexbii 3103	. . . . . . . . 9 ⊢ (∃𝑔 ∈ 𝐵 ∃ℎ ∈ 𝐶 𝑥 = ((𝑓𝐺𝑔)𝐺ℎ) ↔ ∃𝑔 ∈ 𝐵 ∃𝑡(𝑡 = (𝑓𝐺𝑔) ∧ ∃ℎ ∈ 𝐶 𝑥 = (𝑡𝐺ℎ)))
57		rexcom4 3283	. . . . . . . . 9 ⊢ (∃𝑔 ∈ 𝐵 ∃𝑡(𝑡 = (𝑓𝐺𝑔) ∧ ∃ℎ ∈ 𝐶 𝑥 = (𝑡𝐺ℎ)) ↔ ∃𝑡∃𝑔 ∈ 𝐵 (𝑡 = (𝑓𝐺𝑔) ∧ ∃ℎ ∈ 𝐶 𝑥 = (𝑡𝐺ℎ)))
58	56, 57	bitri 277	. . . . . . . 8 ⊢ (∃𝑔 ∈ 𝐵 ∃ℎ ∈ 𝐶 𝑥 = ((𝑓𝐺𝑔)𝐺ℎ) ↔ ∃𝑡∃𝑔 ∈ 𝐵 (𝑡 = (𝑓𝐺𝑔) ∧ ∃ℎ ∈ 𝐶 𝑥 = (𝑡𝐺ℎ)))
59	58	rexbii 3103	. . . . . . 7 ⊢ (∃𝑓 ∈ 𝐴 ∃𝑔 ∈ 𝐵 ∃ℎ ∈ 𝐶 𝑥 = ((𝑓𝐺𝑔)𝐺ℎ) ↔ ∃𝑓 ∈ 𝐴 ∃𝑡∃𝑔 ∈ 𝐵 (𝑡 = (𝑓𝐺𝑔) ∧ ∃ℎ ∈ 𝐶 𝑥 = (𝑡𝐺ℎ)))
60		rexcom4 3283	. . . . . . 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 3226	. . . . . . . 8 ⊢ (∃𝑓 ∈ 𝐴 ∃𝑔 ∈ 𝐵 (𝑡 = (𝑓𝐺𝑔) ∧ ∃ℎ ∈ 𝐶 𝑥 = (𝑡𝐺ℎ)) ↔ (∃𝑓 ∈ 𝐴 ∃𝑔 ∈ 𝐵 𝑡 = (𝑓𝐺𝑔) ∧ ∃ℎ ∈ 𝐶 𝑥 = (𝑡𝐺ℎ)))
62	61	exbii 1862	. . . . . . 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 299	. . . . . 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 316	. . . . 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 281	. . . 4 ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P ∧ 𝐶 ∈ P) → (𝑥 ∈ ((𝐴𝐹𝐵)𝐹𝐶) ↔ ∃𝑓 ∈ 𝐴 ∃𝑔 ∈ 𝐵 ∃ℎ ∈ 𝐶 𝑥 = ((𝑓𝐺𝑔)𝐺ℎ)))
66	32, 37, 65	3bitr4rd 314	. . 3 ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P ∧ 𝐶 ∈ P) → (𝑥 ∈ ((𝐴𝐹𝐵)𝐹𝐶) ↔ 𝑥 ∈ (𝐴𝐹(𝐵𝐹𝐶))))
67	66	eqrdv 2754	. 2 ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P ∧ 𝐶 ∈ P) → ((𝐴𝐹𝐵)𝐹𝐶) = (𝐴𝐹(𝐵𝐹𝐶)))
68		genpass.4	. . 3 ⊢ dom 𝐹 = (P × P)
69		0npr 10940	. . 3 ⊢ ¬ ∅ ∈ P
70	68, 69	ndmovass 7573	. 2 ⊢ (¬ (𝐴 ∈ P ∧ 𝐵 ∈ P ∧ 𝐶 ∈ P) → ((𝐴𝐹𝐵)𝐹𝐶) = (𝐴𝐹(𝐵𝐹𝐶)))
71	67, 70	pm2.61i 183	1 ⊢ ((𝐴𝐹𝐵)𝐹𝐶) = (𝐴𝐹(𝐵𝐹𝐶))

Syntax hints:   → wi 4   ↔ wb 208   ∧ wa 398   ∧ w3a 1095   = wceq 1554  ∃wex 1793   ∈ wcel 2136  {cab 2734  ∃wrex 3080   × cxp 5638  dom cdm 5640  (class class class)co 7385   ∈ cmpo 7387  Qcnq 10800  Pcnp 10807

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1809  ax-4 1823  ax-5 1924  ax-6 1981  ax-7 2022  ax-8 2138  ax-9 2146  ax-10 2169  ax-11 2185  ax-12 2206  ax-ext 2728  ax-sep 5240  ax-nul 5250  ax-pow 5316  ax-pr 5384  ax-un 7707  ax-inf2 9586

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 857  df-3or 1096  df-3an 1097  df-tru 1557  df-fal 1567  df-ex 1794  df-nf 1798  df-sb 2085  df-mo 2560  df-eu 2590  df-clab 2735  df-cleq 2748  df-clel 2831  df-nfc 2905  df-ne 2952  df-ral 3071  df-rex 3081  df-rab 3409  df-v 3450  df-sbc 3740  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-pss 3919  df-nul 4281  df-if 4475  df-pw 4551  df-sn 4577  df-pr 4579  df-op 4583  df-uni 4860  df-br 5095  df-opab 5157  df-tr 5202  df-id 5535  df-eprel 5540  df-po 5548  df-so 5549  df-fr 5593  df-we 5595  df-xp 5646  df-rel 5647  df-cnv 5648  df-co 5649  df-dm 5650  df-ord 6338  df-on 6339  df-lim 6340  df-suc 6341  df-iota 6466  df-fun 6512  df-fv 6518  df-ov 7388  df-oprab 7389  df-mpo 7390  df-om 7836  df-ni 10820  df-nq 10860  df-np 10929

This theorem is referenced by:  addasspr  10970  mulasspr  10972