Theorem th3qlem1 6615

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 1927	. . . 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 581	. . . . . . 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 2233	. . . . . . . . . . . . 13 ⊢ (𝐴 = [𝑦] ∼ → (𝐴 ∈ (𝑆 / ∼ ) ↔ [𝑦] ∼ ∈ (𝑆 / ∼ )))
4		eleq1 2233	. . . . . . . . . . . . 13 ⊢ (𝐵 = [𝑧] ∼ → (𝐵 ∈ (𝑆 / ∼ ) ↔ [𝑧] ∼ ∈ (𝑆 / ∼ )))
5	3, 4	bi2anan9 601	. . . . . . . . . . . 12 ⊢ ((𝐴 = [𝑦] ∼ ∧ 𝐵 = [𝑧] ∼ ) → ((𝐴 ∈ (𝑆 / ∼ ) ∧ 𝐵 ∈ (𝑆 / ∼ )) ↔ ([𝑦] ∼ ∈ (𝑆 / ∼ ) ∧ [𝑧] ∼ ∈ (𝑆 / ∼ ))))
6	5	adantr 274	. . . . . . . . . . 11 ⊢ (((𝐴 = [𝑦] ∼ ∧ 𝐵 = [𝑧] ∼ ) ∧ (𝐴 = [𝑤] ∼ ∧ 𝐵 = [𝑣] ∼ )) → ((𝐴 ∈ (𝑆 / ∼ ) ∧ 𝐵 ∈ (𝑆 / ∼ )) ↔ ([𝑦] ∼ ∈ (𝑆 / ∼ ) ∧ [𝑧] ∼ ∈ (𝑆 / ∼ ))))
7	6	biimpac 296	. . . . . . . . . 10 ⊢ (((𝐴 ∈ (𝑆 / ∼ ) ∧ 𝐵 ∈ (𝑆 / ∼ )) ∧ ((𝐴 = [𝑦] ∼ ∧ 𝐵 = [𝑧] ∼ ) ∧ (𝐴 = [𝑤] ∼ ∧ 𝐵 = [𝑣] ∼ ))) → ([𝑦] ∼ ∈ (𝑆 / ∼ ) ∧ [𝑧] ∼ ∈ (𝑆 / ∼ )))
8		eqtr2 2189	. . . . . . . . . . . . 13 ⊢ ((𝐴 = [𝑦] ∼ ∧ 𝐴 = [𝑤] ∼ ) → [𝑦] ∼ = [𝑤] ∼ )
9		eqtr2 2189	. . . . . . . . . . . . 13 ⊢ ((𝐵 = [𝑧] ∼ ∧ 𝐵 = [𝑣] ∼ ) → [𝑧] ∼ = [𝑣] ∼ )
10	8, 9	anim12i 336	. . . . . . . . . . . 12 ⊢ (((𝐴 = [𝑦] ∼ ∧ 𝐴 = [𝑤] ∼ ) ∧ (𝐵 = [𝑧] ∼ ∧ 𝐵 = [𝑣] ∼ )) → ([𝑦] ∼ = [𝑤] ∼ ∧ [𝑧] ∼ = [𝑣] ∼ ))
11	10	an4s 583	. . . . . . . . . . 11 ⊢ (((𝐴 = [𝑦] ∼ ∧ 𝐵 = [𝑧] ∼ ) ∧ (𝐴 = [𝑤] ∼ ∧ 𝐵 = [𝑣] ∼ )) → ([𝑦] ∼ = [𝑤] ∼ ∧ [𝑧] ∼ = [𝑣] ∼ ))
12	11	adantl 275	. . . . . . . . . 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 526	. . . . . . . . . . . . 13 ⊢ ((([𝑦] ∼ ∈ (𝑆 / ∼ ) ∧ [𝑧] ∼ ∈ (𝑆 / ∼ )) ∧ ([𝑦] ∼ = [𝑤] ∼ ∧ [𝑧] ∼ = [𝑣] ∼ )) → [𝑦] ∼ = [𝑤] ∼ )
16		erdm 6523	. . . . . . . . . . . . . . . 16 ⊢ ( ∼ Er 𝑆 → dom ∼ = 𝑆)
17	13, 16	ax-mp 5	. . . . . . . . . . . . . . 15 ⊢ dom ∼ = 𝑆
18		simpll 524	. . . . . . . . . . . . . . 15 ⊢ ((([𝑦] ∼ ∈ (𝑆 / ∼ ) ∧ [𝑧] ∼ ∈ (𝑆 / ∼ )) ∧ ([𝑦] ∼ = [𝑤] ∼ ∧ [𝑧] ∼ = [𝑣] ∼ )) → [𝑦] ∼ ∈ (𝑆 / ∼ ))
19		ecelqsdm 6583	. . . . . . . . . . . . . . 15 ⊢ ((dom ∼ = 𝑆 ∧ [𝑦] ∼ ∈ (𝑆 / ∼ )) → 𝑦 ∈ 𝑆)
20	17, 18, 19	sylancr 412	. . . . . . . . . . . . . 14 ⊢ ((([𝑦] ∼ ∈ (𝑆 / ∼ ) ∧ [𝑧] ∼ ∈ (𝑆 / ∼ )) ∧ ([𝑦] ∼ = [𝑤] ∼ ∧ [𝑧] ∼ = [𝑣] ∼ )) → 𝑦 ∈ 𝑆)
21	14, 20	erth 6557	. . . . . . . . . . . . 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 527	. . . . . . . . . . . . 13 ⊢ ((([𝑦] ∼ ∈ (𝑆 / ∼ ) ∧ [𝑧] ∼ ∈ (𝑆 / ∼ )) ∧ ([𝑦] ∼ = [𝑤] ∼ ∧ [𝑧] ∼ = [𝑣] ∼ )) → [𝑧] ∼ = [𝑣] ∼ )
24		simplr 525	. . . . . . . . . . . . . . 15 ⊢ ((([𝑦] ∼ ∈ (𝑆 / ∼ ) ∧ [𝑧] ∼ ∈ (𝑆 / ∼ )) ∧ ([𝑦] ∼ = [𝑤] ∼ ∧ [𝑧] ∼ = [𝑣] ∼ )) → [𝑧] ∼ ∈ (𝑆 / ∼ ))
25		ecelqsdm 6583	. . . . . . . . . . . . . . 15 ⊢ ((dom ∼ = 𝑆 ∧ [𝑧] ∼ ∈ (𝑆 / ∼ )) → 𝑧 ∈ 𝑆)
26	17, 24, 25	sylancr 412	. . . . . . . . . . . . . 14 ⊢ ((([𝑦] ∼ ∈ (𝑆 / ∼ ) ∧ [𝑧] ∼ ∈ (𝑆 / ∼ )) ∧ ([𝑦] ∼ = [𝑤] ∼ ∧ [𝑧] ∼ = [𝑣] ∼ )) → 𝑧 ∈ 𝑆)
27	14, 26	erth 6557	. . . . . . . . . . . . 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 2248	. . . . . . . . . . . . . 14 ⊢ ((([𝑦] ∼ ∈ (𝑆 / ∼ ) ∧ [𝑧] ∼ ∈ (𝑆 / ∼ )) ∧ ([𝑦] ∼ = [𝑤] ∼ ∧ [𝑧] ∼ = [𝑣] ∼ )) → [𝑤] ∼ ∈ (𝑆 / ∼ ))
30		ecelqsdm 6583	. . . . . . . . . . . . . 14 ⊢ ((dom ∼ = 𝑆 ∧ [𝑤] ∼ ∈ (𝑆 / ∼ )) → 𝑤 ∈ 𝑆)
31	17, 29, 30	sylancr 412	. . . . . . . . . . . . 13 ⊢ ((([𝑦] ∼ ∈ (𝑆 / ∼ ) ∧ [𝑧] ∼ ∈ (𝑆 / ∼ )) ∧ ([𝑦] ∼ = [𝑤] ∼ ∧ [𝑧] ∼ = [𝑣] ∼ )) → 𝑤 ∈ 𝑆)
32	23, 24	eqeltrrd 2248	. . . . . . . . . . . . . 14 ⊢ ((([𝑦] ∼ ∈ (𝑆 / ∼ ) ∧ [𝑧] ∼ ∈ (𝑆 / ∼ )) ∧ ([𝑦] ∼ = [𝑤] ∼ ∧ [𝑧] ∼ = [𝑣] ∼ )) → [𝑣] ∼ ∈ (𝑆 / ∼ ))
33		ecelqsdm 6583	. . . . . . . . . . . . . 14 ⊢ ((dom ∼ = 𝑆 ∧ [𝑣] ∼ ∈ (𝑆 / ∼ )) → 𝑣 ∈ 𝑆)
34	17, 32, 33	sylancr 412	. . . . . . . . . . . . 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 1234	. . . . . . . . . . . 12 ⊢ ((([𝑦] ∼ ∈ (𝑆 / ∼ ) ∧ [𝑧] ∼ ∈ (𝑆 / ∼ )) ∧ ([𝑦] ∼ = [𝑤] ∼ ∧ [𝑧] ∼ = [𝑣] ∼ )) → ((𝑦 ∼ 𝑤 ∧ 𝑧 ∼ 𝑣) → (𝑦 + 𝑧) ∼ (𝑤 + 𝑣)))
37	22, 28, 36	mp2and 431	. . . . . . . . . . 11 ⊢ ((([𝑦] ∼ ∈ (𝑆 / ∼ ) ∧ [𝑧] ∼ ∈ (𝑆 / ∼ )) ∧ ([𝑦] ∼ = [𝑤] ∼ ∧ [𝑧] ∼ = [𝑣] ∼ )) → (𝑦 + 𝑧) ∼ (𝑤 + 𝑣))
38	14, 37	erthi 6559	. . . . . . . . . 10 ⊢ ((([𝑦] ∼ ∈ (𝑆 / ∼ ) ∧ [𝑧] ∼ ∈ (𝑆 / ∼ )) ∧ ([𝑦] ∼ = [𝑤] ∼ ∧ [𝑧] ∼ = [𝑣] ∼ )) → [(𝑦 + 𝑧)] ∼ = [(𝑤 + 𝑣)] ∼ )
39	7, 12, 38	syl2anc 409	. . . . . . . . 9 ⊢ (((𝐴 ∈ (𝑆 / ∼ ) ∧ 𝐵 ∈ (𝑆 / ∼ )) ∧ ((𝐴 = [𝑦] ∼ ∧ 𝐵 = [𝑧] ∼ ) ∧ (𝐴 = [𝑤] ∼ ∧ 𝐵 = [𝑣] ∼ ))) → [(𝑦 + 𝑧)] ∼ = [(𝑤 + 𝑣)] ∼ )
40		eqeq12 2183	. . . . . . . . 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 361	. . . . . . 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 1890	. . . . 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 1890	. . . 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 1868	. 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 2177	. . . . . 6 ⊢ (𝑥 = 𝑢 → (𝑥 = [(𝑦 + 𝑧)] ∼ ↔ 𝑢 = [(𝑦 + 𝑧)] ∼ ))
49	48	anbi2d 461	. . . . 5 ⊢ (𝑥 = 𝑢 → (((𝐴 = [𝑦] ∼ ∧ 𝐵 = [𝑧] ∼ ) ∧ 𝑥 = [(𝑦 + 𝑧)] ∼ ) ↔ ((𝐴 = [𝑦] ∼ ∧ 𝐵 = [𝑧] ∼ ) ∧ 𝑢 = [(𝑦 + 𝑧)] ∼ )))
50	49	2exbidv 1861	. . . 4 ⊢ (𝑥 = 𝑢 → (∃𝑦∃𝑧((𝐴 = [𝑦] ∼ ∧ 𝐵 = [𝑧] ∼ ) ∧ 𝑥 = [(𝑦 + 𝑧)] ∼ ) ↔ ∃𝑦∃𝑧((𝐴 = [𝑦] ∼ ∧ 𝐵 = [𝑧] ∼ ) ∧ 𝑢 = [(𝑦 + 𝑧)] ∼ )))
51		eceq1 6548	. . . . . . . 8 ⊢ (𝑦 = 𝑤 → [𝑦] ∼ = [𝑤] ∼ )
52	51	eqeq2d 2182	. . . . . . 7 ⊢ (𝑦 = 𝑤 → (𝐴 = [𝑦] ∼ ↔ 𝐴 = [𝑤] ∼ ))
53		eceq1 6548	. . . . . . . 8 ⊢ (𝑧 = 𝑣 → [𝑧] ∼ = [𝑣] ∼ )
54	53	eqeq2d 2182	. . . . . . 7 ⊢ (𝑧 = 𝑣 → (𝐵 = [𝑧] ∼ ↔ 𝐵 = [𝑣] ∼ ))
55	52, 54	bi2anan9 601	. . . . . 6 ⊢ ((𝑦 = 𝑤 ∧ 𝑧 = 𝑣) → ((𝐴 = [𝑦] ∼ ∧ 𝐵 = [𝑧] ∼ ) ↔ (𝐴 = [𝑤] ∼ ∧ 𝐵 = [𝑣] ∼ )))
56		oveq12 5862	. . . . . . . 8 ⊢ ((𝑦 = 𝑤 ∧ 𝑧 = 𝑣) → (𝑦 + 𝑧) = (𝑤 + 𝑣))
57	56	eceq1d 6549	. . . . . . 7 ⊢ ((𝑦 = 𝑤 ∧ 𝑧 = 𝑣) → [(𝑦 + 𝑧)] ∼ = [(𝑤 + 𝑣)] ∼ )
58	57	eqeq2d 2182	. . . . . 6 ⊢ ((𝑦 = 𝑤 ∧ 𝑧 = 𝑣) → (𝑢 = [(𝑦 + 𝑧)] ∼ ↔ 𝑢 = [(𝑤 + 𝑣)] ∼ ))
59	55, 58	anbi12d 470	. . . . 5 ⊢ ((𝑦 = 𝑤 ∧ 𝑧 = 𝑣) → (((𝐴 = [𝑦] ∼ ∧ 𝐵 = [𝑧] ∼ ) ∧ 𝑢 = [(𝑦 + 𝑧)] ∼ ) ↔ ((𝐴 = [𝑤] ∼ ∧ 𝐵 = [𝑣] ∼ ) ∧ 𝑢 = [(𝑤 + 𝑣)] ∼ )))
60	59	cbvex2v 1917	. . . 4 ⊢ (∃𝑦∃𝑧((𝐴 = [𝑦] ∼ ∧ 𝐵 = [𝑧] ∼ ) ∧ 𝑢 = [(𝑦 + 𝑧)] ∼ ) ↔ ∃𝑤∃𝑣((𝐴 = [𝑤] ∼ ∧ 𝐵 = [𝑣] ∼ ) ∧ 𝑢 = [(𝑤 + 𝑣)] ∼ ))
61	50, 60	bitrdi 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 2080	. 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 1346   = wceq 1348  ∃wex 1485  ∃*wmo 2020   ∈ wcel 2141   class class class wbr 3989  dom cdm 4611  (class class class)co 5853   Er wer 6510  [cec 6511   / cqs 6512

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-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-14 2144  ax-ext 2152  ax-sep 4107  ax-pow 4160  ax-pr 4194

This theorem depends on definitions:  df-bi 116  df-3an 975  df-tru 1351  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ral 2453  df-rex 2454  df-v 2732  df-sbc 2956  df-un 3125  df-in 3127  df-ss 3134  df-pw 3568  df-sn 3589  df-pr 3590  df-op 3592  df-uni 3797  df-br 3990  df-opab 4051  df-xp 4617  df-rel 4618  df-cnv 4619  df-co 4620  df-dm 4621  df-rn 4622  df-res 4623  df-ima 4624  df-iota 5160  df-fv 5206  df-ov 5856  df-er 6513  df-ec 6515  df-qs 6519

This theorem is referenced by:  th3qlem2  6616