Theorem reusv3 5367

Description: Two ways to express single-valuedness of a class expression 𝐶(𝑦). See reusv1 5359 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 2846	. . . . 5 ⊢ (𝑦 = 𝑧 → (𝐶 ∈ 𝐴 ↔ 𝐷 ∈ 𝐴))
4	1, 3	anbi12d 644	. . . 4 ⊢ (𝑦 = 𝑧 → ((𝜑 ∧ 𝐶 ∈ 𝐴) ↔ (𝜓 ∧ 𝐷 ∈ 𝐴)))
5	4	cbvrexvw 3242	. . 3 ⊢ (∃𝑦 ∈ 𝐵 (𝜑 ∧ 𝐶 ∈ 𝐴) ↔ ∃𝑧 ∈ 𝐵 (𝜓 ∧ 𝐷 ∈ 𝐴))
6		nfra2w 3299	. . . . 5 ⊢ Ⅎ𝑧∀𝑦 ∈ 𝐵 ∀𝑧 ∈ 𝐵 ((𝜑 ∧ 𝜓) → 𝐶 = 𝐷)
7		nfv 1947	. . . . 5 ⊢ Ⅎ𝑧∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 (𝜑 → 𝑥 = 𝐶)
8	6, 7	nfim 1929	. . . 4 ⊢ Ⅎ𝑧(∀𝑦 ∈ 𝐵 ∀𝑧 ∈ 𝐵 ((𝜑 ∧ 𝜓) → 𝐶 = 𝐷) → ∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 (𝜑 → 𝑥 = 𝐶))
9		risset 3238	. . . . . 6 ⊢ (𝐷 ∈ 𝐴 ↔ ∃𝑥 ∈ 𝐴 𝑥 = 𝐷)
10		ralcom 3291	. . . . . . . . . . . . . 14 ⊢ (∀𝑦 ∈ 𝐵 ∀𝑧 ∈ 𝐵 ((𝜑 ∧ 𝜓) → 𝐶 = 𝐷) ↔ ∀𝑧 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ((𝜑 ∧ 𝜓) → 𝐶 = 𝐷))
11		impexp 456	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ 𝜓) → 𝐶 = 𝐷) ↔ (𝜑 → (𝜓 → 𝐶 = 𝐷)))
12		bi2.04 392	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 → (𝜓 → 𝐶 = 𝐷)) ↔ (𝜓 → (𝜑 → 𝐶 = 𝐷)))
13	11, 12	bitri 278	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝜓) → 𝐶 = 𝐷) ↔ (𝜓 → (𝜑 → 𝐶 = 𝐷)))
14	13	ralbii 3109	. . . . . . . . . . . . . . . 16 ⊢ (∀𝑦 ∈ 𝐵 ((𝜑 ∧ 𝜓) → 𝐶 = 𝐷) ↔ ∀𝑦 ∈ 𝐵 (𝜓 → (𝜑 → 𝐶 = 𝐷)))
15		r19.21v 3188	. . . . . . . . . . . . . . . 16 ⊢ (∀𝑦 ∈ 𝐵 (𝜓 → (𝜑 → 𝐶 = 𝐷)) ↔ (𝜓 → ∀𝑦 ∈ 𝐵 (𝜑 → 𝐶 = 𝐷)))
16	14, 15	bitri 278	. . . . . . . . . . . . . . 15 ⊢ (∀𝑦 ∈ 𝐵 ((𝜑 ∧ 𝜓) → 𝐶 = 𝐷) ↔ (𝜓 → ∀𝑦 ∈ 𝐵 (𝜑 → 𝐶 = 𝐷)))
17	16	ralbii 3109	. . . . . . . . . . . . . 14 ⊢ (∀𝑧 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ((𝜑 ∧ 𝜓) → 𝐶 = 𝐷) ↔ ∀𝑧 ∈ 𝐵 (𝜓 → ∀𝑦 ∈ 𝐵 (𝜑 → 𝐶 = 𝐷)))
18	10, 17	bitri 278	. . . . . . . . . . . . 13 ⊢ (∀𝑦 ∈ 𝐵 ∀𝑧 ∈ 𝐵 ((𝜑 ∧ 𝜓) → 𝐶 = 𝐷) ↔ ∀𝑧 ∈ 𝐵 (𝜓 → ∀𝑦 ∈ 𝐵 (𝜑 → 𝐶 = 𝐷)))
19		rsp 3251	. . . . . . . . . . . . 13 ⊢ (∀𝑧 ∈ 𝐵 (𝜓 → ∀𝑦 ∈ 𝐵 (𝜑 → 𝐶 = 𝐷)) → (𝑧 ∈ 𝐵 → (𝜓 → ∀𝑦 ∈ 𝐵 (𝜑 → 𝐶 = 𝐷))))
20	18, 19	sylbi 220	. . . . . . . . . . . 12 ⊢ (∀𝑦 ∈ 𝐵 ∀𝑧 ∈ 𝐵 ((𝜑 ∧ 𝜓) → 𝐶 = 𝐷) → (𝑧 ∈ 𝐵 → (𝜓 → ∀𝑦 ∈ 𝐵 (𝜑 → 𝐶 = 𝐷))))
21	20	com3l 90	. . . . . . . . . . 11 ⊢ (𝑧 ∈ 𝐵 → (𝜓 → (∀𝑦 ∈ 𝐵 ∀𝑧 ∈ 𝐵 ((𝜑 ∧ 𝜓) → 𝐶 = 𝐷) → ∀𝑦 ∈ 𝐵 (𝜑 → 𝐶 = 𝐷))))
22	21	imp31 423	. . . . . . . . . 10 ⊢ (((𝑧 ∈ 𝐵 ∧ 𝜓) ∧ ∀𝑦 ∈ 𝐵 ∀𝑧 ∈ 𝐵 ((𝜑 ∧ 𝜓) → 𝐶 = 𝐷)) → ∀𝑦 ∈ 𝐵 (𝜑 → 𝐶 = 𝐷))
23		eqeq1 2765	. . . . . . . . . . . . 13 ⊢ (𝑥 = 𝐷 → (𝑥 = 𝐶 ↔ 𝐷 = 𝐶))
24		eqcom 2768	. . . . . . . . . . . . 13 ⊢ (𝐷 = 𝐶 ↔ 𝐶 = 𝐷)
25	23, 24	bitrdi 290	. . . . . . . . . . . 12 ⊢ (𝑥 = 𝐷 → (𝑥 = 𝐶 ↔ 𝐶 = 𝐷))
26	25	imbi2d 343	. . . . . . . . . . 11 ⊢ (𝑥 = 𝐷 → ((𝜑 → 𝑥 = 𝐶) ↔ (𝜑 → 𝐶 = 𝐷)))
27	26	ralbidv 3186	. . . . . . . . . 10 ⊢ (𝑥 = 𝐷 → (∀𝑦 ∈ 𝐵 (𝜑 → 𝑥 = 𝐶) ↔ ∀𝑦 ∈ 𝐵 (𝜑 → 𝐶 = 𝐷)))
28	22, 27	syl5ibrcom 250	. . . . . . . . 9 ⊢ (((𝑧 ∈ 𝐵 ∧ 𝜓) ∧ ∀𝑦 ∈ 𝐵 ∀𝑧 ∈ 𝐵 ((𝜑 ∧ 𝜓) → 𝐶 = 𝐷)) → (𝑥 = 𝐷 → ∀𝑦 ∈ 𝐵 (𝜑 → 𝑥 = 𝐶)))
29	28	reximdv 3178	. . . . . . . 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 418	. . . . . . 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 87	. . . . . 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	biimtrid 245	. . . . 5 ⊢ ((𝑧 ∈ 𝐵 ∧ 𝜓) → (𝐷 ∈ 𝐴 → (∀𝑦 ∈ 𝐵 ∀𝑧 ∈ 𝐵 ((𝜑 ∧ 𝜓) → 𝐶 = 𝐷) → ∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 (𝜑 → 𝑥 = 𝐶))))
33	32	expimpd 459	. . . 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 3263	. . 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 220	. 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 5366	. 2 ⊢ (∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 (𝜑 → 𝑥 = 𝐶) → ∀𝑦 ∈ 𝐵 ∀𝑧 ∈ 𝐵 ((𝜑 ∧ 𝜓) → 𝐶 = 𝐷))
37	35, 36	impbid1 228	1 ⊢ (∃𝑦 ∈ 𝐵 (𝜑 ∧ 𝐶 ∈ 𝐴) → (∀𝑦 ∈ 𝐵 ∀𝑧 ∈ 𝐵 ((𝜑 ∧ 𝜓) → 𝐶 = 𝐷) ↔ ∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 (𝜑 → 𝑥 = 𝐶)))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   = wceq 1570   ∈ wcel 2145  ∀wral 3077  ∃wrex 3087

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 2733

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-clab 2740  df-cleq 2753  df-clel 2836  df-ne 2957  df-ral 3078  df-rex 3088  df-dif 3902  df-nul 4280

This theorem is used by:  cdleme25b  41391  cdleme29b  41412  cdlemk28-3  41945  dihlsscpre  42271  mapdh9a  42826  mapdh9aOLDN  42827