Theorem reusv3 4876

Description: Two ways to express single-valuedness of a class expression 𝐶(𝑦). See reusv1 4866 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 2686	. . . . 5 ⊢ (𝑦 = 𝑧 → (𝐶 ∈ 𝐴 ↔ 𝐷 ∈ 𝐴))
4	1, 3	anbi12d 747	. . . 4 ⊢ (𝑦 = 𝑧 → ((𝜑 ∧ 𝐶 ∈ 𝐴) ↔ (𝜓 ∧ 𝐷 ∈ 𝐴)))
5	4	cbvrexv 3172	. . 3 ⊢ (∃𝑦 ∈ 𝐵 (𝜑 ∧ 𝐶 ∈ 𝐴) ↔ ∃𝑧 ∈ 𝐵 (𝜓 ∧ 𝐷 ∈ 𝐴))
6		nfra2 2946	. . . . 5 ⊢ Ⅎ𝑧∀𝑦 ∈ 𝐵 ∀𝑧 ∈ 𝐵 ((𝜑 ∧ 𝜓) → 𝐶 = 𝐷)
7		nfv 1843	. . . . 5 ⊢ Ⅎ𝑧∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 (𝜑 → 𝑥 = 𝐶)
8	6, 7	nfim 1825	. . . 4 ⊢ Ⅎ𝑧(∀𝑦 ∈ 𝐵 ∀𝑧 ∈ 𝐵 ((𝜑 ∧ 𝜓) → 𝐶 = 𝐷) → ∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 (𝜑 → 𝑥 = 𝐶))
9		risset 3062	. . . . . 6 ⊢ (𝐷 ∈ 𝐴 ↔ ∃𝑥 ∈ 𝐴 𝑥 = 𝐷)
10		ralcom 3098	. . . . . . . . . . . . . 14 ⊢ (∀𝑦 ∈ 𝐵 ∀𝑧 ∈ 𝐵 ((𝜑 ∧ 𝜓) → 𝐶 = 𝐷) ↔ ∀𝑧 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ((𝜑 ∧ 𝜓) → 𝐶 = 𝐷))
11		impexp 462	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ 𝜓) → 𝐶 = 𝐷) ↔ (𝜑 → (𝜓 → 𝐶 = 𝐷)))
12		bi2.04 376	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 → (𝜓 → 𝐶 = 𝐷)) ↔ (𝜓 → (𝜑 → 𝐶 = 𝐷)))
13	11, 12	bitri 264	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝜓) → 𝐶 = 𝐷) ↔ (𝜓 → (𝜑 → 𝐶 = 𝐷)))
14	13	ralbii 2980	. . . . . . . . . . . . . . . 16 ⊢ (∀𝑦 ∈ 𝐵 ((𝜑 ∧ 𝜓) → 𝐶 = 𝐷) ↔ ∀𝑦 ∈ 𝐵 (𝜓 → (𝜑 → 𝐶 = 𝐷)))
15		r19.21v 2960	. . . . . . . . . . . . . . . 16 ⊢ (∀𝑦 ∈ 𝐵 (𝜓 → (𝜑 → 𝐶 = 𝐷)) ↔ (𝜓 → ∀𝑦 ∈ 𝐵 (𝜑 → 𝐶 = 𝐷)))
16	14, 15	bitri 264	. . . . . . . . . . . . . . 15 ⊢ (∀𝑦 ∈ 𝐵 ((𝜑 ∧ 𝜓) → 𝐶 = 𝐷) ↔ (𝜓 → ∀𝑦 ∈ 𝐵 (𝜑 → 𝐶 = 𝐷)))
17	16	ralbii 2980	. . . . . . . . . . . . . 14 ⊢ (∀𝑧 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ((𝜑 ∧ 𝜓) → 𝐶 = 𝐷) ↔ ∀𝑧 ∈ 𝐵 (𝜓 → ∀𝑦 ∈ 𝐵 (𝜑 → 𝐶 = 𝐷)))
18	10, 17	bitri 264	. . . . . . . . . . . . 13 ⊢ (∀𝑦 ∈ 𝐵 ∀𝑧 ∈ 𝐵 ((𝜑 ∧ 𝜓) → 𝐶 = 𝐷) ↔ ∀𝑧 ∈ 𝐵 (𝜓 → ∀𝑦 ∈ 𝐵 (𝜑 → 𝐶 = 𝐷)))
19		rsp 2929	. . . . . . . . . . . . 13 ⊢ (∀𝑧 ∈ 𝐵 (𝜓 → ∀𝑦 ∈ 𝐵 (𝜑 → 𝐶 = 𝐷)) → (𝑧 ∈ 𝐵 → (𝜓 → ∀𝑦 ∈ 𝐵 (𝜑 → 𝐶 = 𝐷))))
20	18, 19	sylbi 207	. . . . . . . . . . . 12 ⊢ (∀𝑦 ∈ 𝐵 ∀𝑧 ∈ 𝐵 ((𝜑 ∧ 𝜓) → 𝐶 = 𝐷) → (𝑧 ∈ 𝐵 → (𝜓 → ∀𝑦 ∈ 𝐵 (𝜑 → 𝐶 = 𝐷))))
21	20	com3l 89	. . . . . . . . . . 11 ⊢ (𝑧 ∈ 𝐵 → (𝜓 → (∀𝑦 ∈ 𝐵 ∀𝑧 ∈ 𝐵 ((𝜑 ∧ 𝜓) → 𝐶 = 𝐷) → ∀𝑦 ∈ 𝐵 (𝜑 → 𝐶 = 𝐷))))
22	21	imp31 448	. . . . . . . . . 10 ⊢ (((𝑧 ∈ 𝐵 ∧ 𝜓) ∧ ∀𝑦 ∈ 𝐵 ∀𝑧 ∈ 𝐵 ((𝜑 ∧ 𝜓) → 𝐶 = 𝐷)) → ∀𝑦 ∈ 𝐵 (𝜑 → 𝐶 = 𝐷))
23		eqeq1 2626	. . . . . . . . . . . . 13 ⊢ (𝑥 = 𝐷 → (𝑥 = 𝐶 ↔ 𝐷 = 𝐶))
24		eqcom 2629	. . . . . . . . . . . . 13 ⊢ (𝐷 = 𝐶 ↔ 𝐶 = 𝐷)
25	23, 24	syl6bb 276	. . . . . . . . . . . 12 ⊢ (𝑥 = 𝐷 → (𝑥 = 𝐶 ↔ 𝐶 = 𝐷))
26	25	imbi2d 330	. . . . . . . . . . 11 ⊢ (𝑥 = 𝐷 → ((𝜑 → 𝑥 = 𝐶) ↔ (𝜑 → 𝐶 = 𝐷)))
27	26	ralbidv 2986	. . . . . . . . . 10 ⊢ (𝑥 = 𝐷 → (∀𝑦 ∈ 𝐵 (𝜑 → 𝑥 = 𝐶) ↔ ∀𝑦 ∈ 𝐵 (𝜑 → 𝐶 = 𝐷)))
28	22, 27	syl5ibrcom 237	. . . . . . . . 9 ⊢ (((𝑧 ∈ 𝐵 ∧ 𝜓) ∧ ∀𝑦 ∈ 𝐵 ∀𝑧 ∈ 𝐵 ((𝜑 ∧ 𝜓) → 𝐶 = 𝐷)) → (𝑥 = 𝐷 → ∀𝑦 ∈ 𝐵 (𝜑 → 𝑥 = 𝐶)))
29	28	reximdv 3016	. . . . . . . 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 450	. . . . . . 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 86	. . . . . 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 232	. . . . 5 ⊢ ((𝑧 ∈ 𝐵 ∧ 𝜓) → (𝐷 ∈ 𝐴 → (∀𝑦 ∈ 𝐵 ∀𝑧 ∈ 𝐵 ((𝜑 ∧ 𝜓) → 𝐶 = 𝐷) → ∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 (𝜑 → 𝑥 = 𝐶))))
33	32	expimpd 629	. . . 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 3024	. . 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 207	. 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 4875	. 2 ⊢ (∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 (𝜑 → 𝑥 = 𝐶) → ∀𝑦 ∈ 𝐵 ∀𝑧 ∈ 𝐵 ((𝜑 ∧ 𝜓) → 𝐶 = 𝐷))
37	35, 36	impbid1 215	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   ↔ wb 196   ∧ wa 384   = wceq 1483   ∈ wcel 1990  ∀wral 2912  ∃wrex 2913

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-v 3202  df-dif 3577  df-nul 3916

This theorem is referenced by:  cdleme25b  35642  cdleme29b  35663  cdlemk28-3  36196  dihlsscpre  36523  mapdh9a  37079  mapdh9aOLDN  37080