Theorem reusv3 4445

Description: Two ways to express single-valuedness of a class expression 𝐶(𝑦). See reusv1 4443 for the connection to uniqueness. (Contributed by NM, 27-Dec-2012.)

Hypotheses

Ref	Expression
reusv3.1	⊢ (𝑦 = 𝑧 → (𝜑 ↔ 𝜓))
reusv3.2	⊢ (𝑦 = 𝑧 → 𝐶 = 𝐷)

Assertion

Ref	Expression
reusv3	⊢ (∃𝑦 ∈ 𝐵 (𝜑 ∧ 𝐶 ∈ 𝐴) → (∀𝑦 ∈ 𝐵 ∀𝑧 ∈ 𝐵 ((𝜑 ∧ 𝜓) → 𝐶 = 𝐷) ↔ ∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 (𝜑 → 𝑥 = 𝐶)))

Distinct variable groups:   𝑥,𝑦,𝑧,𝐵   𝑥,𝐶,𝑧   𝑥,𝐷,𝑦   𝜑,𝑥,𝑧   𝜓,𝑥,𝑦   𝑥,𝐴,𝑦,𝑧

Allowed substitution hints:   𝜑(𝑦)   𝜓(𝑧)   𝐶(𝑦)   𝐷(𝑧)

Proof of Theorem reusv3

Step	Hyp	Ref	Expression
1		reusv3.1	. . . . 5 ⊢ (𝑦 = 𝑧 → (𝜑 ↔ 𝜓))
2		reusv3.2	. . . . . 6 ⊢ (𝑦 = 𝑧 → 𝐶 = 𝐷)
3	2	eleq1d 2239	. . . . 5 ⊢ (𝑦 = 𝑧 → (𝐶 ∈ 𝐴 ↔ 𝐷 ∈ 𝐴))
4	1, 3	anbi12d 470	. . . 4 ⊢ (𝑦 = 𝑧 → ((𝜑 ∧ 𝐶 ∈ 𝐴) ↔ (𝜓 ∧ 𝐷 ∈ 𝐴)))
5	4	cbvrexv 2697	. . 3 ⊢ (∃𝑦 ∈ 𝐵 (𝜑 ∧ 𝐶 ∈ 𝐴) ↔ ∃𝑧 ∈ 𝐵 (𝜓 ∧ 𝐷 ∈ 𝐴))
6		nfra2xy 2512	. . . . 5 ⊢ Ⅎ𝑧∀𝑦 ∈ 𝐵 ∀𝑧 ∈ 𝐵 ((𝜑 ∧ 𝜓) → 𝐶 = 𝐷)
7		nfv 1521	. . . . 5 ⊢ Ⅎ𝑧∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 (𝜑 → 𝑥 = 𝐶)
8	6, 7	nfim 1565	. . . 4 ⊢ Ⅎ𝑧(∀𝑦 ∈ 𝐵 ∀𝑧 ∈ 𝐵 ((𝜑 ∧ 𝜓) → 𝐶 = 𝐷) → ∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 (𝜑 → 𝑥 = 𝐶))
9		risset 2498	. . . . . 6 ⊢ (𝐷 ∈ 𝐴 ↔ ∃𝑥 ∈ 𝐴 𝑥 = 𝐷)
10		ralcom 2633	. . . . . . . . . . . . . 14 ⊢ (∀𝑦 ∈ 𝐵 ∀𝑧 ∈ 𝐵 ((𝜑 ∧ 𝜓) → 𝐶 = 𝐷) ↔ ∀𝑧 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ((𝜑 ∧ 𝜓) → 𝐶 = 𝐷))
11		impexp 261	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ 𝜓) → 𝐶 = 𝐷) ↔ (𝜑 → (𝜓 → 𝐶 = 𝐷)))
12		bi2.04 247	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 → (𝜓 → 𝐶 = 𝐷)) ↔ (𝜓 → (𝜑 → 𝐶 = 𝐷)))
13	11, 12	bitri 183	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝜓) → 𝐶 = 𝐷) ↔ (𝜓 → (𝜑 → 𝐶 = 𝐷)))
14	13	ralbii 2476	. . . . . . . . . . . . . . . 16 ⊢ (∀𝑦 ∈ 𝐵 ((𝜑 ∧ 𝜓) → 𝐶 = 𝐷) ↔ ∀𝑦 ∈ 𝐵 (𝜓 → (𝜑 → 𝐶 = 𝐷)))
15		r19.21v 2547	. . . . . . . . . . . . . . . 16 ⊢ (∀𝑦 ∈ 𝐵 (𝜓 → (𝜑 → 𝐶 = 𝐷)) ↔ (𝜓 → ∀𝑦 ∈ 𝐵 (𝜑 → 𝐶 = 𝐷)))
16	14, 15	bitri 183	. . . . . . . . . . . . . . 15 ⊢ (∀𝑦 ∈ 𝐵 ((𝜑 ∧ 𝜓) → 𝐶 = 𝐷) ↔ (𝜓 → ∀𝑦 ∈ 𝐵 (𝜑 → 𝐶 = 𝐷)))
17	16	ralbii 2476	. . . . . . . . . . . . . 14 ⊢ (∀𝑧 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ((𝜑 ∧ 𝜓) → 𝐶 = 𝐷) ↔ ∀𝑧 ∈ 𝐵 (𝜓 → ∀𝑦 ∈ 𝐵 (𝜑 → 𝐶 = 𝐷)))
18	10, 17	bitri 183	. . . . . . . . . . . . 13 ⊢ (∀𝑦 ∈ 𝐵 ∀𝑧 ∈ 𝐵 ((𝜑 ∧ 𝜓) → 𝐶 = 𝐷) ↔ ∀𝑧 ∈ 𝐵 (𝜓 → ∀𝑦 ∈ 𝐵 (𝜑 → 𝐶 = 𝐷)))
19		rsp 2517	. . . . . . . . . . . . 13 ⊢ (∀𝑧 ∈ 𝐵 (𝜓 → ∀𝑦 ∈ 𝐵 (𝜑 → 𝐶 = 𝐷)) → (𝑧 ∈ 𝐵 → (𝜓 → ∀𝑦 ∈ 𝐵 (𝜑 → 𝐶 = 𝐷))))
20	18, 19	sylbi 120	. . . . . . . . . . . 12 ⊢ (∀𝑦 ∈ 𝐵 ∀𝑧 ∈ 𝐵 ((𝜑 ∧ 𝜓) → 𝐶 = 𝐷) → (𝑧 ∈ 𝐵 → (𝜓 → ∀𝑦 ∈ 𝐵 (𝜑 → 𝐶 = 𝐷))))
21	20	com3l 81	. . . . . . . . . . 11 ⊢ (𝑧 ∈ 𝐵 → (𝜓 → (∀𝑦 ∈ 𝐵 ∀𝑧 ∈ 𝐵 ((𝜑 ∧ 𝜓) → 𝐶 = 𝐷) → ∀𝑦 ∈ 𝐵 (𝜑 → 𝐶 = 𝐷))))
22	21	imp31 254	. . . . . . . . . 10 ⊢ (((𝑧 ∈ 𝐵 ∧ 𝜓) ∧ ∀𝑦 ∈ 𝐵 ∀𝑧 ∈ 𝐵 ((𝜑 ∧ 𝜓) → 𝐶 = 𝐷)) → ∀𝑦 ∈ 𝐵 (𝜑 → 𝐶 = 𝐷))
23		eqeq1 2177	. . . . . . . . . . . . 13 ⊢ (𝑥 = 𝐷 → (𝑥 = 𝐶 ↔ 𝐷 = 𝐶))
24		eqcom 2172	. . . . . . . . . . . . 13 ⊢ (𝐷 = 𝐶 ↔ 𝐶 = 𝐷)
25	23, 24	bitrdi 195	. . . . . . . . . . . 12 ⊢ (𝑥 = 𝐷 → (𝑥 = 𝐶 ↔ 𝐶 = 𝐷))
26	25	imbi2d 229	. . . . . . . . . . 11 ⊢ (𝑥 = 𝐷 → ((𝜑 → 𝑥 = 𝐶) ↔ (𝜑 → 𝐶 = 𝐷)))
27	26	ralbidv 2470	. . . . . . . . . 10 ⊢ (𝑥 = 𝐷 → (∀𝑦 ∈ 𝐵 (𝜑 → 𝑥 = 𝐶) ↔ ∀𝑦 ∈ 𝐵 (𝜑 → 𝐶 = 𝐷)))
28	22, 27	syl5ibrcom 156	. . . . . . . . 9 ⊢ (((𝑧 ∈ 𝐵 ∧ 𝜓) ∧ ∀𝑦 ∈ 𝐵 ∀𝑧 ∈ 𝐵 ((𝜑 ∧ 𝜓) → 𝐶 = 𝐷)) → (𝑥 = 𝐷 → ∀𝑦 ∈ 𝐵 (𝜑 → 𝑥 = 𝐶)))
29	28	reximdv 2571	. . . . . . . 8 ⊢ (((𝑧 ∈ 𝐵 ∧ 𝜓) ∧ ∀𝑦 ∈ 𝐵 ∀𝑧 ∈ 𝐵 ((𝜑 ∧ 𝜓) → 𝐶 = 𝐷)) → (∃𝑥 ∈ 𝐴 𝑥 = 𝐷 → ∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 (𝜑 → 𝑥 = 𝐶)))
30	29	ex 114	. . . . . . 7 ⊢ ((𝑧 ∈ 𝐵 ∧ 𝜓) → (∀𝑦 ∈ 𝐵 ∀𝑧 ∈ 𝐵 ((𝜑 ∧ 𝜓) → 𝐶 = 𝐷) → (∃𝑥 ∈ 𝐴 𝑥 = 𝐷 → ∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 (𝜑 → 𝑥 = 𝐶))))
31	30	com23 78	. . . . . 6 ⊢ ((𝑧 ∈ 𝐵 ∧ 𝜓) → (∃𝑥 ∈ 𝐴 𝑥 = 𝐷 → (∀𝑦 ∈ 𝐵 ∀𝑧 ∈ 𝐵 ((𝜑 ∧ 𝜓) → 𝐶 = 𝐷) → ∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 (𝜑 → 𝑥 = 𝐶))))
32	9, 31	syl5bi 151	. . . . 5 ⊢ ((𝑧 ∈ 𝐵 ∧ 𝜓) → (𝐷 ∈ 𝐴 → (∀𝑦 ∈ 𝐵 ∀𝑧 ∈ 𝐵 ((𝜑 ∧ 𝜓) → 𝐶 = 𝐷) → ∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 (𝜑 → 𝑥 = 𝐶))))
33	32	expimpd 361	. . . 4 ⊢ (𝑧 ∈ 𝐵 → ((𝜓 ∧ 𝐷 ∈ 𝐴) → (∀𝑦 ∈ 𝐵 ∀𝑧 ∈ 𝐵 ((𝜑 ∧ 𝜓) → 𝐶 = 𝐷) → ∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 (𝜑 → 𝑥 = 𝐶))))
34	8, 33	rexlimi 2580	. . 3 ⊢ (∃𝑧 ∈ 𝐵 (𝜓 ∧ 𝐷 ∈ 𝐴) → (∀𝑦 ∈ 𝐵 ∀𝑧 ∈ 𝐵 ((𝜑 ∧ 𝜓) → 𝐶 = 𝐷) → ∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 (𝜑 → 𝑥 = 𝐶)))
35	5, 34	sylbi 120	. 2 ⊢ (∃𝑦 ∈ 𝐵 (𝜑 ∧ 𝐶 ∈ 𝐴) → (∀𝑦 ∈ 𝐵 ∀𝑧 ∈ 𝐵 ((𝜑 ∧ 𝜓) → 𝐶 = 𝐷) → ∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 (𝜑 → 𝑥 = 𝐶)))
36	1, 2	reusv3i 4444	. 2 ⊢ (∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 (𝜑 → 𝑥 = 𝐶) → ∀𝑦 ∈ 𝐵 ∀𝑧 ∈ 𝐵 ((𝜑 ∧ 𝜓) → 𝐶 = 𝐷))
37	35, 36	impbid1 141	1 ⊢ (∃𝑦 ∈ 𝐵 (𝜑 ∧ 𝐶 ∈ 𝐴) → (∀𝑦 ∈ 𝐵 ∀𝑧 ∈ 𝐵 ((𝜑 ∧ 𝜓) → 𝐶 = 𝐷) ↔ ∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 (𝜑 → 𝑥 = 𝐶)))

Syntax hints:   → wi 4   ∧ wa 103   ↔ wb 104   = wceq 1348   ∈ wcel 2141  ∀wral 2448  ∃wrex 2449

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-ext 2152

This theorem depends on definitions:  df-bi 116  df-tru 1351  df-nf 1454  df-sb 1756  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ral 2453  df-rex 2454

This theorem is referenced by: (None)