Theorem th3qlem1 6408

Description: Lemma for Exercise 44 version of Theorem 3Q of [Enderton] p. 60. The third hypothesis is the compatibility assumption. (Contributed by NM, 3-Aug-1995.) (Revised by Mario Carneiro, 9-Jul-2014.)

Hypotheses

Ref	Expression
th3qlem1.1	⊢ ∼ Er 𝑆
th3qlem1.3	⊢ (((𝑦 ∈ 𝑆 ∧ 𝑤 ∈ 𝑆) ∧ (𝑧 ∈ 𝑆 ∧ 𝑣 ∈ 𝑆)) → ((𝑦 ∼ 𝑤 ∧ 𝑧 ∼ 𝑣) → (𝑦 + 𝑧) ∼ (𝑤 + 𝑣)))

Assertion

Ref	Expression
th3qlem1	⊢ ((𝐴 ∈ (𝑆 / ∼ ) ∧ 𝐵 ∈ (𝑆 / ∼ )) → ∃*𝑥∃𝑦∃𝑧((𝐴 = [𝑦] ∼ ∧ 𝐵 = [𝑧] ∼ ) ∧ 𝑥 = [(𝑦 + 𝑧)] ∼ ))

Distinct variable groups:   𝑥,𝑦,𝑧,𝑤,𝑣, +   𝑥, ∼ ,𝑦,𝑧,𝑤,𝑣   𝑥,𝑆,𝑦,𝑧,𝑤,𝑣   𝑥,𝐴,𝑦,𝑧,𝑤,𝑣   𝑥,𝐵,𝑦,𝑧,𝑤,𝑣

Proof of Theorem th3qlem1

Dummy variable 𝑢 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		ee4anv 1858	. . . 4 ⊢ (∃𝑦∃𝑧∃𝑤∃𝑣(((𝐴 = [𝑦] ∼ ∧ 𝐵 = [𝑧] ∼ ) ∧ 𝑥 = [(𝑦 + 𝑧)] ∼ ) ∧ ((𝐴 = [𝑤] ∼ ∧ 𝐵 = [𝑣] ∼ ) ∧ 𝑢 = [(𝑤 + 𝑣)] ∼ )) ↔ (∃𝑦∃𝑧((𝐴 = [𝑦] ∼ ∧ 𝐵 = [𝑧] ∼ ) ∧ 𝑥 = [(𝑦 + 𝑧)] ∼ ) ∧ ∃𝑤∃𝑣((𝐴 = [𝑤] ∼ ∧ 𝐵 = [𝑣] ∼ ) ∧ 𝑢 = [(𝑤 + 𝑣)] ∼ )))
2		an4 554	. . . . . . 7 ⊢ ((((𝐴 = [𝑦] ∼ ∧ 𝐵 = [𝑧] ∼ ) ∧ 𝑥 = [(𝑦 + 𝑧)] ∼ ) ∧ ((𝐴 = [𝑤] ∼ ∧ 𝐵 = [𝑣] ∼ ) ∧ 𝑢 = [(𝑤 + 𝑣)] ∼ )) ↔ (((𝐴 = [𝑦] ∼ ∧ 𝐵 = [𝑧] ∼ ) ∧ (𝐴 = [𝑤] ∼ ∧ 𝐵 = [𝑣] ∼ )) ∧ (𝑥 = [(𝑦 + 𝑧)] ∼ ∧ 𝑢 = [(𝑤 + 𝑣)] ∼ )))
3		eleq1 2151	. . . . . . . . . . . . 13 ⊢ (𝐴 = [𝑦] ∼ → (𝐴 ∈ (𝑆 / ∼ ) ↔ [𝑦] ∼ ∈ (𝑆 / ∼ )))
4		eleq1 2151	. . . . . . . . . . . . 13 ⊢ (𝐵 = [𝑧] ∼ → (𝐵 ∈ (𝑆 / ∼ ) ↔ [𝑧] ∼ ∈ (𝑆 / ∼ )))
5	3, 4	bi2anan9 574	. . . . . . . . . . . 12 ⊢ ((𝐴 = [𝑦] ∼ ∧ 𝐵 = [𝑧] ∼ ) → ((𝐴 ∈ (𝑆 / ∼ ) ∧ 𝐵 ∈ (𝑆 / ∼ )) ↔ ([𝑦] ∼ ∈ (𝑆 / ∼ ) ∧ [𝑧] ∼ ∈ (𝑆 / ∼ ))))
6	5	adantr 271	. . . . . . . . . . 11 ⊢ (((𝐴 = [𝑦] ∼ ∧ 𝐵 = [𝑧] ∼ ) ∧ (𝐴 = [𝑤] ∼ ∧ 𝐵 = [𝑣] ∼ )) → ((𝐴 ∈ (𝑆 / ∼ ) ∧ 𝐵 ∈ (𝑆 / ∼ )) ↔ ([𝑦] ∼ ∈ (𝑆 / ∼ ) ∧ [𝑧] ∼ ∈ (𝑆 / ∼ ))))
7	6	biimpac 293	. . . . . . . . . 10 ⊢ (((𝐴 ∈ (𝑆 / ∼ ) ∧ 𝐵 ∈ (𝑆 / ∼ )) ∧ ((𝐴 = [𝑦] ∼ ∧ 𝐵 = [𝑧] ∼ ) ∧ (𝐴 = [𝑤] ∼ ∧ 𝐵 = [𝑣] ∼ ))) → ([𝑦] ∼ ∈ (𝑆 / ∼ ) ∧ [𝑧] ∼ ∈ (𝑆 / ∼ )))
8		eqtr2 2107	. . . . . . . . . . . . 13 ⊢ ((𝐴 = [𝑦] ∼ ∧ 𝐴 = [𝑤] ∼ ) → [𝑦] ∼ = [𝑤] ∼ )
9		eqtr2 2107	. . . . . . . . . . . . 13 ⊢ ((𝐵 = [𝑧] ∼ ∧ 𝐵 = [𝑣] ∼ ) → [𝑧] ∼ = [𝑣] ∼ )
10	8, 9	anim12i 332	. . . . . . . . . . . 12 ⊢ (((𝐴 = [𝑦] ∼ ∧ 𝐴 = [𝑤] ∼ ) ∧ (𝐵 = [𝑧] ∼ ∧ 𝐵 = [𝑣] ∼ )) → ([𝑦] ∼ = [𝑤] ∼ ∧ [𝑧] ∼ = [𝑣] ∼ ))
11	10	an4s 556	. . . . . . . . . . 11 ⊢ (((𝐴 = [𝑦] ∼ ∧ 𝐵 = [𝑧] ∼ ) ∧ (𝐴 = [𝑤] ∼ ∧ 𝐵 = [𝑣] ∼ )) → ([𝑦] ∼ = [𝑤] ∼ ∧ [𝑧] ∼ = [𝑣] ∼ ))
12	11	adantl 272	. . . . . . . . . 10 ⊢ (((𝐴 ∈ (𝑆 / ∼ ) ∧ 𝐵 ∈ (𝑆 / ∼ )) ∧ ((𝐴 = [𝑦] ∼ ∧ 𝐵 = [𝑧] ∼ ) ∧ (𝐴 = [𝑤] ∼ ∧ 𝐵 = [𝑣] ∼ ))) → ([𝑦] ∼ = [𝑤] ∼ ∧ [𝑧] ∼ = [𝑣] ∼ ))
13		th3qlem1.1	. . . . . . . . . . . 12 ⊢ ∼ Er 𝑆
14	13	a1i 9	. . . . . . . . . . 11 ⊢ ((([𝑦] ∼ ∈ (𝑆 / ∼ ) ∧ [𝑧] ∼ ∈ (𝑆 / ∼ )) ∧ ([𝑦] ∼ = [𝑤] ∼ ∧ [𝑧] ∼ = [𝑣] ∼ )) → ∼ Er 𝑆)
15		simprl 499	. . . . . . . . . . . . 13 ⊢ ((([𝑦] ∼ ∈ (𝑆 / ∼ ) ∧ [𝑧] ∼ ∈ (𝑆 / ∼ )) ∧ ([𝑦] ∼ = [𝑤] ∼ ∧ [𝑧] ∼ = [𝑣] ∼ )) → [𝑦] ∼ = [𝑤] ∼ )
16		erdm 6316	. . . . . . . . . . . . . . . 16 ⊢ ( ∼ Er 𝑆 → dom ∼ = 𝑆)
17	13, 16	ax-mp 7	. . . . . . . . . . . . . . 15 ⊢ dom ∼ = 𝑆
18		simpll 497	. . . . . . . . . . . . . . 15 ⊢ ((([𝑦] ∼ ∈ (𝑆 / ∼ ) ∧ [𝑧] ∼ ∈ (𝑆 / ∼ )) ∧ ([𝑦] ∼ = [𝑤] ∼ ∧ [𝑧] ∼ = [𝑣] ∼ )) → [𝑦] ∼ ∈ (𝑆 / ∼ ))
19		ecelqsdm 6376	. . . . . . . . . . . . . . 15 ⊢ ((dom ∼ = 𝑆 ∧ [𝑦] ∼ ∈ (𝑆 / ∼ )) → 𝑦 ∈ 𝑆)
20	17, 18, 19	sylancr 406	. . . . . . . . . . . . . 14 ⊢ ((([𝑦] ∼ ∈ (𝑆 / ∼ ) ∧ [𝑧] ∼ ∈ (𝑆 / ∼ )) ∧ ([𝑦] ∼ = [𝑤] ∼ ∧ [𝑧] ∼ = [𝑣] ∼ )) → 𝑦 ∈ 𝑆)
21	14, 20	erth 6350	. . . . . . . . . . . . 13 ⊢ ((([𝑦] ∼ ∈ (𝑆 / ∼ ) ∧ [𝑧] ∼ ∈ (𝑆 / ∼ )) ∧ ([𝑦] ∼ = [𝑤] ∼ ∧ [𝑧] ∼ = [𝑣] ∼ )) → (𝑦 ∼ 𝑤 ↔ [𝑦] ∼ = [𝑤] ∼ ))
22	15, 21	mpbird 166	. . . . . . . . . . . 12 ⊢ ((([𝑦] ∼ ∈ (𝑆 / ∼ ) ∧ [𝑧] ∼ ∈ (𝑆 / ∼ )) ∧ ([𝑦] ∼ = [𝑤] ∼ ∧ [𝑧] ∼ = [𝑣] ∼ )) → 𝑦 ∼ 𝑤)
23		simprr 500	. . . . . . . . . . . . 13 ⊢ ((([𝑦] ∼ ∈ (𝑆 / ∼ ) ∧ [𝑧] ∼ ∈ (𝑆 / ∼ )) ∧ ([𝑦] ∼ = [𝑤] ∼ ∧ [𝑧] ∼ = [𝑣] ∼ )) → [𝑧] ∼ = [𝑣] ∼ )
24		simplr 498	. . . . . . . . . . . . . . 15 ⊢ ((([𝑦] ∼ ∈ (𝑆 / ∼ ) ∧ [𝑧] ∼ ∈ (𝑆 / ∼ )) ∧ ([𝑦] ∼ = [𝑤] ∼ ∧ [𝑧] ∼ = [𝑣] ∼ )) → [𝑧] ∼ ∈ (𝑆 / ∼ ))
25		ecelqsdm 6376	. . . . . . . . . . . . . . 15 ⊢ ((dom ∼ = 𝑆 ∧ [𝑧] ∼ ∈ (𝑆 / ∼ )) → 𝑧 ∈ 𝑆)
26	17, 24, 25	sylancr 406	. . . . . . . . . . . . . 14 ⊢ ((([𝑦] ∼ ∈ (𝑆 / ∼ ) ∧ [𝑧] ∼ ∈ (𝑆 / ∼ )) ∧ ([𝑦] ∼ = [𝑤] ∼ ∧ [𝑧] ∼ = [𝑣] ∼ )) → 𝑧 ∈ 𝑆)
27	14, 26	erth 6350	. . . . . . . . . . . . 13 ⊢ ((([𝑦] ∼ ∈ (𝑆 / ∼ ) ∧ [𝑧] ∼ ∈ (𝑆 / ∼ )) ∧ ([𝑦] ∼ = [𝑤] ∼ ∧ [𝑧] ∼ = [𝑣] ∼ )) → (𝑧 ∼ 𝑣 ↔ [𝑧] ∼ = [𝑣] ∼ ))
28	23, 27	mpbird 166	. . . . . . . . . . . 12 ⊢ ((([𝑦] ∼ ∈ (𝑆 / ∼ ) ∧ [𝑧] ∼ ∈ (𝑆 / ∼ )) ∧ ([𝑦] ∼ = [𝑤] ∼ ∧ [𝑧] ∼ = [𝑣] ∼ )) → 𝑧 ∼ 𝑣)
29	15, 18	eqeltrrd 2166	. . . . . . . . . . . . . 14 ⊢ ((([𝑦] ∼ ∈ (𝑆 / ∼ ) ∧ [𝑧] ∼ ∈ (𝑆 / ∼ )) ∧ ([𝑦] ∼ = [𝑤] ∼ ∧ [𝑧] ∼ = [𝑣] ∼ )) → [𝑤] ∼ ∈ (𝑆 / ∼ ))
30		ecelqsdm 6376	. . . . . . . . . . . . . 14 ⊢ ((dom ∼ = 𝑆 ∧ [𝑤] ∼ ∈ (𝑆 / ∼ )) → 𝑤 ∈ 𝑆)
31	17, 29, 30	sylancr 406	. . . . . . . . . . . . 13 ⊢ ((([𝑦] ∼ ∈ (𝑆 / ∼ ) ∧ [𝑧] ∼ ∈ (𝑆 / ∼ )) ∧ ([𝑦] ∼ = [𝑤] ∼ ∧ [𝑧] ∼ = [𝑣] ∼ )) → 𝑤 ∈ 𝑆)
32	23, 24	eqeltrrd 2166	. . . . . . . . . . . . . 14 ⊢ ((([𝑦] ∼ ∈ (𝑆 / ∼ ) ∧ [𝑧] ∼ ∈ (𝑆 / ∼ )) ∧ ([𝑦] ∼ = [𝑤] ∼ ∧ [𝑧] ∼ = [𝑣] ∼ )) → [𝑣] ∼ ∈ (𝑆 / ∼ ))
33		ecelqsdm 6376	. . . . . . . . . . . . . 14 ⊢ ((dom ∼ = 𝑆 ∧ [𝑣] ∼ ∈ (𝑆 / ∼ )) → 𝑣 ∈ 𝑆)
34	17, 32, 33	sylancr 406	. . . . . . . . . . . . 13 ⊢ ((([𝑦] ∼ ∈ (𝑆 / ∼ ) ∧ [𝑧] ∼ ∈ (𝑆 / ∼ )) ∧ ([𝑦] ∼ = [𝑤] ∼ ∧ [𝑧] ∼ = [𝑣] ∼ )) → 𝑣 ∈ 𝑆)
35		th3qlem1.3	. . . . . . . . . . . . 13 ⊢ (((𝑦 ∈ 𝑆 ∧ 𝑤 ∈ 𝑆) ∧ (𝑧 ∈ 𝑆 ∧ 𝑣 ∈ 𝑆)) → ((𝑦 ∼ 𝑤 ∧ 𝑧 ∼ 𝑣) → (𝑦 + 𝑧) ∼ (𝑤 + 𝑣)))
36	20, 31, 26, 34, 35	syl22anc 1176	. . . . . . . . . . . 12 ⊢ ((([𝑦] ∼ ∈ (𝑆 / ∼ ) ∧ [𝑧] ∼ ∈ (𝑆 / ∼ )) ∧ ([𝑦] ∼ = [𝑤] ∼ ∧ [𝑧] ∼ = [𝑣] ∼ )) → ((𝑦 ∼ 𝑤 ∧ 𝑧 ∼ 𝑣) → (𝑦 + 𝑧) ∼ (𝑤 + 𝑣)))
37	22, 28, 36	mp2and 425	. . . . . . . . . . 11 ⊢ ((([𝑦] ∼ ∈ (𝑆 / ∼ ) ∧ [𝑧] ∼ ∈ (𝑆 / ∼ )) ∧ ([𝑦] ∼ = [𝑤] ∼ ∧ [𝑧] ∼ = [𝑣] ∼ )) → (𝑦 + 𝑧) ∼ (𝑤 + 𝑣))
38	14, 37	erthi 6352	. . . . . . . . . 10 ⊢ ((([𝑦] ∼ ∈ (𝑆 / ∼ ) ∧ [𝑧] ∼ ∈ (𝑆 / ∼ )) ∧ ([𝑦] ∼ = [𝑤] ∼ ∧ [𝑧] ∼ = [𝑣] ∼ )) → [(𝑦 + 𝑧)] ∼ = [(𝑤 + 𝑣)] ∼ )
39	7, 12, 38	syl2anc 404	. . . . . . . . 9 ⊢ (((𝐴 ∈ (𝑆 / ∼ ) ∧ 𝐵 ∈ (𝑆 / ∼ )) ∧ ((𝐴 = [𝑦] ∼ ∧ 𝐵 = [𝑧] ∼ ) ∧ (𝐴 = [𝑤] ∼ ∧ 𝐵 = [𝑣] ∼ ))) → [(𝑦 + 𝑧)] ∼ = [(𝑤 + 𝑣)] ∼ )
40		eqeq12 2101	. . . . . . . . 9 ⊢ ((𝑥 = [(𝑦 + 𝑧)] ∼ ∧ 𝑢 = [(𝑤 + 𝑣)] ∼ ) → (𝑥 = 𝑢 ↔ [(𝑦 + 𝑧)] ∼ = [(𝑤 + 𝑣)] ∼ ))
41	39, 40	syl5ibrcom 156	. . . . . . . 8 ⊢ (((𝐴 ∈ (𝑆 / ∼ ) ∧ 𝐵 ∈ (𝑆 / ∼ )) ∧ ((𝐴 = [𝑦] ∼ ∧ 𝐵 = [𝑧] ∼ ) ∧ (𝐴 = [𝑤] ∼ ∧ 𝐵 = [𝑣] ∼ ))) → ((𝑥 = [(𝑦 + 𝑧)] ∼ ∧ 𝑢 = [(𝑤 + 𝑣)] ∼ ) → 𝑥 = 𝑢))
42	41	expimpd 356	. . . . . . 7 ⊢ ((𝐴 ∈ (𝑆 / ∼ ) ∧ 𝐵 ∈ (𝑆 / ∼ )) → ((((𝐴 = [𝑦] ∼ ∧ 𝐵 = [𝑧] ∼ ) ∧ (𝐴 = [𝑤] ∼ ∧ 𝐵 = [𝑣] ∼ )) ∧ (𝑥 = [(𝑦 + 𝑧)] ∼ ∧ 𝑢 = [(𝑤 + 𝑣)] ∼ )) → 𝑥 = 𝑢))
43	2, 42	syl5bi 151	. . . . . 6 ⊢ ((𝐴 ∈ (𝑆 / ∼ ) ∧ 𝐵 ∈ (𝑆 / ∼ )) → ((((𝐴 = [𝑦] ∼ ∧ 𝐵 = [𝑧] ∼ ) ∧ 𝑥 = [(𝑦 + 𝑧)] ∼ ) ∧ ((𝐴 = [𝑤] ∼ ∧ 𝐵 = [𝑣] ∼ ) ∧ 𝑢 = [(𝑤 + 𝑣)] ∼ )) → 𝑥 = 𝑢))
44	43	exlimdvv 1826	. . . . 5 ⊢ ((𝐴 ∈ (𝑆 / ∼ ) ∧ 𝐵 ∈ (𝑆 / ∼ )) → (∃𝑤∃𝑣(((𝐴 = [𝑦] ∼ ∧ 𝐵 = [𝑧] ∼ ) ∧ 𝑥 = [(𝑦 + 𝑧)] ∼ ) ∧ ((𝐴 = [𝑤] ∼ ∧ 𝐵 = [𝑣] ∼ ) ∧ 𝑢 = [(𝑤 + 𝑣)] ∼ )) → 𝑥 = 𝑢))
45	44	exlimdvv 1826	. . . 4 ⊢ ((𝐴 ∈ (𝑆 / ∼ ) ∧ 𝐵 ∈ (𝑆 / ∼ )) → (∃𝑦∃𝑧∃𝑤∃𝑣(((𝐴 = [𝑦] ∼ ∧ 𝐵 = [𝑧] ∼ ) ∧ 𝑥 = [(𝑦 + 𝑧)] ∼ ) ∧ ((𝐴 = [𝑤] ∼ ∧ 𝐵 = [𝑣] ∼ ) ∧ 𝑢 = [(𝑤 + 𝑣)] ∼ )) → 𝑥 = 𝑢))
46	1, 45	syl5bir 152	. . 3 ⊢ ((𝐴 ∈ (𝑆 / ∼ ) ∧ 𝐵 ∈ (𝑆 / ∼ )) → ((∃𝑦∃𝑧((𝐴 = [𝑦] ∼ ∧ 𝐵 = [𝑧] ∼ ) ∧ 𝑥 = [(𝑦 + 𝑧)] ∼ ) ∧ ∃𝑤∃𝑣((𝐴 = [𝑤] ∼ ∧ 𝐵 = [𝑣] ∼ ) ∧ 𝑢 = [(𝑤 + 𝑣)] ∼ )) → 𝑥 = 𝑢))
47	46	alrimivv 1804	. 2 ⊢ ((𝐴 ∈ (𝑆 / ∼ ) ∧ 𝐵 ∈ (𝑆 / ∼ )) → ∀𝑥∀𝑢((∃𝑦∃𝑧((𝐴 = [𝑦] ∼ ∧ 𝐵 = [𝑧] ∼ ) ∧ 𝑥 = [(𝑦 + 𝑧)] ∼ ) ∧ ∃𝑤∃𝑣((𝐴 = [𝑤] ∼ ∧ 𝐵 = [𝑣] ∼ ) ∧ 𝑢 = [(𝑤 + 𝑣)] ∼ )) → 𝑥 = 𝑢))
48		eqeq1 2095	. . . . . 6 ⊢ (𝑥 = 𝑢 → (𝑥 = [(𝑦 + 𝑧)] ∼ ↔ 𝑢 = [(𝑦 + 𝑧)] ∼ ))
49	48	anbi2d 453	. . . . 5 ⊢ (𝑥 = 𝑢 → (((𝐴 = [𝑦] ∼ ∧ 𝐵 = [𝑧] ∼ ) ∧ 𝑥 = [(𝑦 + 𝑧)] ∼ ) ↔ ((𝐴 = [𝑦] ∼ ∧ 𝐵 = [𝑧] ∼ ) ∧ 𝑢 = [(𝑦 + 𝑧)] ∼ )))
50	49	2exbidv 1797	. . . 4 ⊢ (𝑥 = 𝑢 → (∃𝑦∃𝑧((𝐴 = [𝑦] ∼ ∧ 𝐵 = [𝑧] ∼ ) ∧ 𝑥 = [(𝑦 + 𝑧)] ∼ ) ↔ ∃𝑦∃𝑧((𝐴 = [𝑦] ∼ ∧ 𝐵 = [𝑧] ∼ ) ∧ 𝑢 = [(𝑦 + 𝑧)] ∼ )))
51		eceq1 6341	. . . . . . . 8 ⊢ (𝑦 = 𝑤 → [𝑦] ∼ = [𝑤] ∼ )
52	51	eqeq2d 2100	. . . . . . 7 ⊢ (𝑦 = 𝑤 → (𝐴 = [𝑦] ∼ ↔ 𝐴 = [𝑤] ∼ ))
53		eceq1 6341	. . . . . . . 8 ⊢ (𝑧 = 𝑣 → [𝑧] ∼ = [𝑣] ∼ )
54	53	eqeq2d 2100	. . . . . . 7 ⊢ (𝑧 = 𝑣 → (𝐵 = [𝑧] ∼ ↔ 𝐵 = [𝑣] ∼ ))
55	52, 54	bi2anan9 574	. . . . . 6 ⊢ ((𝑦 = 𝑤 ∧ 𝑧 = 𝑣) → ((𝐴 = [𝑦] ∼ ∧ 𝐵 = [𝑧] ∼ ) ↔ (𝐴 = [𝑤] ∼ ∧ 𝐵 = [𝑣] ∼ )))
56		oveq12 5675	. . . . . . . 8 ⊢ ((𝑦 = 𝑤 ∧ 𝑧 = 𝑣) → (𝑦 + 𝑧) = (𝑤 + 𝑣))
57	56	eceq1d 6342	. . . . . . 7 ⊢ ((𝑦 = 𝑤 ∧ 𝑧 = 𝑣) → [(𝑦 + 𝑧)] ∼ = [(𝑤 + 𝑣)] ∼ )
58	57	eqeq2d 2100	. . . . . 6 ⊢ ((𝑦 = 𝑤 ∧ 𝑧 = 𝑣) → (𝑢 = [(𝑦 + 𝑧)] ∼ ↔ 𝑢 = [(𝑤 + 𝑣)] ∼ ))
59	55, 58	anbi12d 458	. . . . 5 ⊢ ((𝑦 = 𝑤 ∧ 𝑧 = 𝑣) → (((𝐴 = [𝑦] ∼ ∧ 𝐵 = [𝑧] ∼ ) ∧ 𝑢 = [(𝑦 + 𝑧)] ∼ ) ↔ ((𝐴 = [𝑤] ∼ ∧ 𝐵 = [𝑣] ∼ ) ∧ 𝑢 = [(𝑤 + 𝑣)] ∼ )))
60	59	cbvex2v 1848	. . . 4 ⊢ (∃𝑦∃𝑧((𝐴 = [𝑦] ∼ ∧ 𝐵 = [𝑧] ∼ ) ∧ 𝑢 = [(𝑦 + 𝑧)] ∼ ) ↔ ∃𝑤∃𝑣((𝐴 = [𝑤] ∼ ∧ 𝐵 = [𝑣] ∼ ) ∧ 𝑢 = [(𝑤 + 𝑣)] ∼ ))
61	50, 60	syl6bb 195	. . 3 ⊢ (𝑥 = 𝑢 → (∃𝑦∃𝑧((𝐴 = [𝑦] ∼ ∧ 𝐵 = [𝑧] ∼ ) ∧ 𝑥 = [(𝑦 + 𝑧)] ∼ ) ↔ ∃𝑤∃𝑣((𝐴 = [𝑤] ∼ ∧ 𝐵 = [𝑣] ∼ ) ∧ 𝑢 = [(𝑤 + 𝑣)] ∼ )))
62	61	mo4 2010	. 2 ⊢ (∃*𝑥∃𝑦∃𝑧((𝐴 = [𝑦] ∼ ∧ 𝐵 = [𝑧] ∼ ) ∧ 𝑥 = [(𝑦 + 𝑧)] ∼ ) ↔ ∀𝑥∀𝑢((∃𝑦∃𝑧((𝐴 = [𝑦] ∼ ∧ 𝐵 = [𝑧] ∼ ) ∧ 𝑥 = [(𝑦 + 𝑧)] ∼ ) ∧ ∃𝑤∃𝑣((𝐴 = [𝑤] ∼ ∧ 𝐵 = [𝑣] ∼ ) ∧ 𝑢 = [(𝑤 + 𝑣)] ∼ )) → 𝑥 = 𝑢))
63	47, 62	sylibr 133	1 ⊢ ((𝐴 ∈ (𝑆 / ∼ ) ∧ 𝐵 ∈ (𝑆 / ∼ )) → ∃*𝑥∃𝑦∃𝑧((𝐴 = [𝑦] ∼ ∧ 𝐵 = [𝑧] ∼ ) ∧ 𝑥 = [(𝑦 + 𝑧)] ∼ ))

Syntax hints:   → wi 4   ∧ wa 103   ↔ wb 104  ∀wal 1288   = wceq 1290  ∃wex 1427   ∈ wcel 1439  ∃*wmo 1950   class class class wbr 3851  dom cdm 4452  (class class class)co 5666   Er wer 6303  [cec 6304   / cqs 6305

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 666  ax-5 1382  ax-7 1383  ax-gen 1384  ax-ie1 1428  ax-ie2 1429  ax-8 1441  ax-10 1442  ax-11 1443  ax-i12 1444  ax-bndl 1445  ax-4 1446  ax-14 1451  ax-17 1465  ax-i9 1469  ax-ial 1473  ax-i5r 1474  ax-ext 2071  ax-sep 3963  ax-pow 4015  ax-pr 4045

This theorem depends on definitions:  df-bi 116  df-3an 927  df-tru 1293  df-nf 1396  df-sb 1694  df-eu 1952  df-mo 1953  df-clab 2076  df-cleq 2082  df-clel 2085  df-nfc 2218  df-ral 2365  df-rex 2366  df-v 2622  df-sbc 2842  df-un 3004  df-in 3006  df-ss 3013  df-pw 3435  df-sn 3456  df-pr 3457  df-op 3459  df-uni 3660  df-br 3852  df-opab 3906  df-xp 4458  df-rel 4459  df-cnv 4460  df-co 4461  df-dm 4462  df-rn 4463  df-res 4464  df-ima 4465  df-iota 4993  df-fv 5036  df-ov 5669  df-er 6306  df-ec 6308  df-qs 6312

This theorem is referenced by:  th3qlem2  6409