Theorem 2reuimp 43671

Description: Implication of a double restricted existential uniqueness in terms of restricted existential quantification and restricted universal quantification if the class of the quantified elements is not empty. (Contributed by AV, 13-Mar-2023.)

Hypotheses

Ref	Expression
2reuimp.c	⊢ (𝑏 = 𝑐 → (𝜑 ↔ 𝜃))
2reuimp.d	⊢ (𝑎 = 𝑑 → (𝜑 ↔ 𝜒))
2reuimp.a	⊢ (𝑎 = 𝑑 → (𝜃 ↔ 𝜏))
2reuimp.e	⊢ (𝑏 = 𝑒 → (𝜑 ↔ 𝜂))
2reuimp.f	⊢ (𝑐 = 𝑓 → (𝜃 ↔ 𝜓))

Assertion

Ref	Expression
2reuimp	⊢ ((𝑉 ≠ ∅ ∧ ∃!𝑎 ∈ 𝑉 ∃!𝑏 ∈ 𝑉 𝜑) → ∃𝑎 ∈ 𝑉 ∀𝑑 ∈ 𝑉 ∀𝑏 ∈ 𝑉 ∃𝑒 ∈ 𝑉 ∀𝑓 ∈ 𝑉 ∃𝑐 ∈ 𝑉 ((𝜒 ∧ (𝜏 → 𝑏 = 𝑐)) → (𝜓 → (𝜂 ∧ (𝑎 = 𝑑 ∧ 𝑒 = 𝑓)))))

Distinct variable groups:   𝑉,𝑎,𝑏,𝑐,𝑑,𝑒,𝑓   𝜑,𝑐,𝑑,𝑒   𝜃,𝑏,𝑑,𝑒,𝑓   𝜒,𝑎,𝑒,𝑓   𝜏,𝑎,𝑒,𝑓   𝜂,𝑏,𝑓   𝜓,𝑐   𝜒,𝑐   𝜂,𝑐

Allowed substitution hints:   𝜑(𝑓,𝑎,𝑏)   𝜓(𝑒,𝑓,𝑎,𝑏,𝑑)   𝜒(𝑏,𝑑)   𝜃(𝑎,𝑐)   𝜏(𝑏,𝑐,𝑑)   𝜂(𝑒,𝑎,𝑑)

Proof of Theorem 2reuimp

Step	Hyp	Ref	Expression
1		r19.28zv 4404	. . . . . . . . . . . 12 ⊢ (𝑉 ≠ ∅ → (∀𝑐 ∈ 𝑉 (𝜒 ∧ (𝜏 → 𝑏 = 𝑐)) ↔ (𝜒 ∧ ∀𝑐 ∈ 𝑉 (𝜏 → 𝑏 = 𝑐))))
2	1	bicomd 226	. . . . . . . . . . 11 ⊢ (𝑉 ≠ ∅ → ((𝜒 ∧ ∀𝑐 ∈ 𝑉 (𝜏 → 𝑏 = 𝑐)) ↔ ∀𝑐 ∈ 𝑉 (𝜒 ∧ (𝜏 → 𝑏 = 𝑐))))
3	2	imbi1d 345	. . . . . . . . . 10 ⊢ (𝑉 ≠ ∅ → (((𝜒 ∧ ∀𝑐 ∈ 𝑉 (𝜏 → 𝑏 = 𝑐)) → 𝑎 = 𝑑) ↔ (∀𝑐 ∈ 𝑉 (𝜒 ∧ (𝜏 → 𝑏 = 𝑐)) → 𝑎 = 𝑑)))
4		r19.36zv 4410	. . . . . . . . . . 11 ⊢ (𝑉 ≠ ∅ → (∃𝑐 ∈ 𝑉 ((𝜒 ∧ (𝜏 → 𝑏 = 𝑐)) → 𝑎 = 𝑑) ↔ (∀𝑐 ∈ 𝑉 (𝜒 ∧ (𝜏 → 𝑏 = 𝑐)) → 𝑎 = 𝑑)))
5		r19.42v 3303	. . . . . . . . . . . . . 14 ⊢ (∃𝑐 ∈ 𝑉 ((𝜂 ∧ (𝜓 → 𝑒 = 𝑓)) ∧ ((𝜒 ∧ (𝜏 → 𝑏 = 𝑐)) → 𝑎 = 𝑑)) ↔ ((𝜂 ∧ (𝜓 → 𝑒 = 𝑓)) ∧ ∃𝑐 ∈ 𝑉 ((𝜒 ∧ (𝜏 → 𝑏 = 𝑐)) → 𝑎 = 𝑑)))
6		pm5.31r 830	. . . . . . . . . . . . . . . 16 ⊢ ((𝜂 ∧ (𝜓 → 𝑒 = 𝑓)) → (𝜓 → (𝜂 ∧ 𝑒 = 𝑓)))
7		pm5.31r 830	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜓 → (𝜂 ∧ 𝑒 = 𝑓)) ∧ ((𝜒 ∧ (𝜏 → 𝑏 = 𝑐)) → 𝑎 = 𝑑)) → ((𝜒 ∧ (𝜏 → 𝑏 = 𝑐)) → ((𝜓 → (𝜂 ∧ 𝑒 = 𝑓)) ∧ 𝑎 = 𝑑)))
8		pm5.31r 830	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑎 = 𝑑 ∧ (𝜓 → (𝜂 ∧ 𝑒 = 𝑓))) → (𝜓 → (𝑎 = 𝑑 ∧ (𝜂 ∧ 𝑒 = 𝑓))))
9		an12 644	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑎 = 𝑑 ∧ (𝜂 ∧ 𝑒 = 𝑓)) ↔ (𝜂 ∧ (𝑎 = 𝑑 ∧ 𝑒 = 𝑓)))
10	8, 9	syl6ib 254	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑎 = 𝑑 ∧ (𝜓 → (𝜂 ∧ 𝑒 = 𝑓))) → (𝜓 → (𝜂 ∧ (𝑎 = 𝑑 ∧ 𝑒 = 𝑓))))
11	10	ancoms 462	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜓 → (𝜂 ∧ 𝑒 = 𝑓)) ∧ 𝑎 = 𝑑) → (𝜓 → (𝜂 ∧ (𝑎 = 𝑑 ∧ 𝑒 = 𝑓))))
12	7, 11	syl6 35	. . . . . . . . . . . . . . . 16 ⊢ (((𝜓 → (𝜂 ∧ 𝑒 = 𝑓)) ∧ ((𝜒 ∧ (𝜏 → 𝑏 = 𝑐)) → 𝑎 = 𝑑)) → ((𝜒 ∧ (𝜏 → 𝑏 = 𝑐)) → (𝜓 → (𝜂 ∧ (𝑎 = 𝑑 ∧ 𝑒 = 𝑓)))))
13	6, 12	sylan 583	. . . . . . . . . . . . . . 15 ⊢ (((𝜂 ∧ (𝜓 → 𝑒 = 𝑓)) ∧ ((𝜒 ∧ (𝜏 → 𝑏 = 𝑐)) → 𝑎 = 𝑑)) → ((𝜒 ∧ (𝜏 → 𝑏 = 𝑐)) → (𝜓 → (𝜂 ∧ (𝑎 = 𝑑 ∧ 𝑒 = 𝑓)))))
14	13	reximi 3206	. . . . . . . . . . . . . 14 ⊢ (∃𝑐 ∈ 𝑉 ((𝜂 ∧ (𝜓 → 𝑒 = 𝑓)) ∧ ((𝜒 ∧ (𝜏 → 𝑏 = 𝑐)) → 𝑎 = 𝑑)) → ∃𝑐 ∈ 𝑉 ((𝜒 ∧ (𝜏 → 𝑏 = 𝑐)) → (𝜓 → (𝜂 ∧ (𝑎 = 𝑑 ∧ 𝑒 = 𝑓)))))
15	5, 14	sylbir 238	. . . . . . . . . . . . 13 ⊢ (((𝜂 ∧ (𝜓 → 𝑒 = 𝑓)) ∧ ∃𝑐 ∈ 𝑉 ((𝜒 ∧ (𝜏 → 𝑏 = 𝑐)) → 𝑎 = 𝑑)) → ∃𝑐 ∈ 𝑉 ((𝜒 ∧ (𝜏 → 𝑏 = 𝑐)) → (𝜓 → (𝜂 ∧ (𝑎 = 𝑑 ∧ 𝑒 = 𝑓)))))
16	15	expcom 417	. . . . . . . . . . . 12 ⊢ (∃𝑐 ∈ 𝑉 ((𝜒 ∧ (𝜏 → 𝑏 = 𝑐)) → 𝑎 = 𝑑) → ((𝜂 ∧ (𝜓 → 𝑒 = 𝑓)) → ∃𝑐 ∈ 𝑉 ((𝜒 ∧ (𝜏 → 𝑏 = 𝑐)) → (𝜓 → (𝜂 ∧ (𝑎 = 𝑑 ∧ 𝑒 = 𝑓))))))
17	16	expd 419	. . . . . . . . . . 11 ⊢ (∃𝑐 ∈ 𝑉 ((𝜒 ∧ (𝜏 → 𝑏 = 𝑐)) → 𝑎 = 𝑑) → (𝜂 → ((𝜓 → 𝑒 = 𝑓) → ∃𝑐 ∈ 𝑉 ((𝜒 ∧ (𝜏 → 𝑏 = 𝑐)) → (𝜓 → (𝜂 ∧ (𝑎 = 𝑑 ∧ 𝑒 = 𝑓)))))))
18	4, 17	syl6bir 257	. . . . . . . . . 10 ⊢ (𝑉 ≠ ∅ → ((∀𝑐 ∈ 𝑉 (𝜒 ∧ (𝜏 → 𝑏 = 𝑐)) → 𝑎 = 𝑑) → (𝜂 → ((𝜓 → 𝑒 = 𝑓) → ∃𝑐 ∈ 𝑉 ((𝜒 ∧ (𝜏 → 𝑏 = 𝑐)) → (𝜓 → (𝜂 ∧ (𝑎 = 𝑑 ∧ 𝑒 = 𝑓))))))))
19	3, 18	sylbid 243	. . . . . . . . 9 ⊢ (𝑉 ≠ ∅ → (((𝜒 ∧ ∀𝑐 ∈ 𝑉 (𝜏 → 𝑏 = 𝑐)) → 𝑎 = 𝑑) → (𝜂 → ((𝜓 → 𝑒 = 𝑓) → ∃𝑐 ∈ 𝑉 ((𝜒 ∧ (𝜏 → 𝑏 = 𝑐)) → (𝜓 → (𝜂 ∧ (𝑎 = 𝑑 ∧ 𝑒 = 𝑓))))))))
20	19	com23 86	. . . . . . . 8 ⊢ (𝑉 ≠ ∅ → (𝜂 → (((𝜒 ∧ ∀𝑐 ∈ 𝑉 (𝜏 → 𝑏 = 𝑐)) → 𝑎 = 𝑑) → ((𝜓 → 𝑒 = 𝑓) → ∃𝑐 ∈ 𝑉 ((𝜒 ∧ (𝜏 → 𝑏 = 𝑐)) → (𝜓 → (𝜂 ∧ (𝑎 = 𝑑 ∧ 𝑒 = 𝑓))))))))
21	20	imp4c 427	. . . . . . 7 ⊢ (𝑉 ≠ ∅ → (((𝜂 ∧ ((𝜒 ∧ ∀𝑐 ∈ 𝑉 (𝜏 → 𝑏 = 𝑐)) → 𝑎 = 𝑑)) ∧ (𝜓 → 𝑒 = 𝑓)) → ∃𝑐 ∈ 𝑉 ((𝜒 ∧ (𝜏 → 𝑏 = 𝑐)) → (𝜓 → (𝜂 ∧ (𝑎 = 𝑑 ∧ 𝑒 = 𝑓))))))
22	21	ralimdv 3145	. . . . . 6 ⊢ (𝑉 ≠ ∅ → (∀𝑓 ∈ 𝑉 ((𝜂 ∧ ((𝜒 ∧ ∀𝑐 ∈ 𝑉 (𝜏 → 𝑏 = 𝑐)) → 𝑎 = 𝑑)) ∧ (𝜓 → 𝑒 = 𝑓)) → ∀𝑓 ∈ 𝑉 ∃𝑐 ∈ 𝑉 ((𝜒 ∧ (𝜏 → 𝑏 = 𝑐)) → (𝜓 → (𝜂 ∧ (𝑎 = 𝑑 ∧ 𝑒 = 𝑓))))))
23	22	reximdv 3232	. . . . 5 ⊢ (𝑉 ≠ ∅ → (∃𝑒 ∈ 𝑉 ∀𝑓 ∈ 𝑉 ((𝜂 ∧ ((𝜒 ∧ ∀𝑐 ∈ 𝑉 (𝜏 → 𝑏 = 𝑐)) → 𝑎 = 𝑑)) ∧ (𝜓 → 𝑒 = 𝑓)) → ∃𝑒 ∈ 𝑉 ∀𝑓 ∈ 𝑉 ∃𝑐 ∈ 𝑉 ((𝜒 ∧ (𝜏 → 𝑏 = 𝑐)) → (𝜓 → (𝜂 ∧ (𝑎 = 𝑑 ∧ 𝑒 = 𝑓))))))
24	23	ralimdv 3145	. . . 4 ⊢ (𝑉 ≠ ∅ → (∀𝑏 ∈ 𝑉 ∃𝑒 ∈ 𝑉 ∀𝑓 ∈ 𝑉 ((𝜂 ∧ ((𝜒 ∧ ∀𝑐 ∈ 𝑉 (𝜏 → 𝑏 = 𝑐)) → 𝑎 = 𝑑)) ∧ (𝜓 → 𝑒 = 𝑓)) → ∀𝑏 ∈ 𝑉 ∃𝑒 ∈ 𝑉 ∀𝑓 ∈ 𝑉 ∃𝑐 ∈ 𝑉 ((𝜒 ∧ (𝜏 → 𝑏 = 𝑐)) → (𝜓 → (𝜂 ∧ (𝑎 = 𝑑 ∧ 𝑒 = 𝑓))))))
25	24	ralimdv 3145	. . 3 ⊢ (𝑉 ≠ ∅ → (∀𝑑 ∈ 𝑉 ∀𝑏 ∈ 𝑉 ∃𝑒 ∈ 𝑉 ∀𝑓 ∈ 𝑉 ((𝜂 ∧ ((𝜒 ∧ ∀𝑐 ∈ 𝑉 (𝜏 → 𝑏 = 𝑐)) → 𝑎 = 𝑑)) ∧ (𝜓 → 𝑒 = 𝑓)) → ∀𝑑 ∈ 𝑉 ∀𝑏 ∈ 𝑉 ∃𝑒 ∈ 𝑉 ∀𝑓 ∈ 𝑉 ∃𝑐 ∈ 𝑉 ((𝜒 ∧ (𝜏 → 𝑏 = 𝑐)) → (𝜓 → (𝜂 ∧ (𝑎 = 𝑑 ∧ 𝑒 = 𝑓))))))
26	25	reximdv 3232	. 2 ⊢ (𝑉 ≠ ∅ → (∃𝑎 ∈ 𝑉 ∀𝑑 ∈ 𝑉 ∀𝑏 ∈ 𝑉 ∃𝑒 ∈ 𝑉 ∀𝑓 ∈ 𝑉 ((𝜂 ∧ ((𝜒 ∧ ∀𝑐 ∈ 𝑉 (𝜏 → 𝑏 = 𝑐)) → 𝑎 = 𝑑)) ∧ (𝜓 → 𝑒 = 𝑓)) → ∃𝑎 ∈ 𝑉 ∀𝑑 ∈ 𝑉 ∀𝑏 ∈ 𝑉 ∃𝑒 ∈ 𝑉 ∀𝑓 ∈ 𝑉 ∃𝑐 ∈ 𝑉 ((𝜒 ∧ (𝜏 → 𝑏 = 𝑐)) → (𝜓 → (𝜂 ∧ (𝑎 = 𝑑 ∧ 𝑒 = 𝑓))))))
27		2reuimp.c	. . 3 ⊢ (𝑏 = 𝑐 → (𝜑 ↔ 𝜃))
28		2reuimp.d	. . 3 ⊢ (𝑎 = 𝑑 → (𝜑 ↔ 𝜒))
29		2reuimp.a	. . 3 ⊢ (𝑎 = 𝑑 → (𝜃 ↔ 𝜏))
30		2reuimp.e	. . 3 ⊢ (𝑏 = 𝑒 → (𝜑 ↔ 𝜂))
31		2reuimp.f	. . 3 ⊢ (𝑐 = 𝑓 → (𝜃 ↔ 𝜓))
32	27, 28, 29, 30, 31	2reuimp0 43670	. 2 ⊢ (∃!𝑎 ∈ 𝑉 ∃!𝑏 ∈ 𝑉 𝜑 → ∃𝑎 ∈ 𝑉 ∀𝑑 ∈ 𝑉 ∀𝑏 ∈ 𝑉 ∃𝑒 ∈ 𝑉 ∀𝑓 ∈ 𝑉 ((𝜂 ∧ ((𝜒 ∧ ∀𝑐 ∈ 𝑉 (𝜏 → 𝑏 = 𝑐)) → 𝑎 = 𝑑)) ∧ (𝜓 → 𝑒 = 𝑓)))
33	26, 32	impel 509	1 ⊢ ((𝑉 ≠ ∅ ∧ ∃!𝑎 ∈ 𝑉 ∃!𝑏 ∈ 𝑉 𝜑) → ∃𝑎 ∈ 𝑉 ∀𝑑 ∈ 𝑉 ∀𝑏 ∈ 𝑉 ∃𝑒 ∈ 𝑉 ∀𝑓 ∈ 𝑉 ∃𝑐 ∈ 𝑉 ((𝜒 ∧ (𝜏 → 𝑏 = 𝑐)) → (𝜓 → (𝜂 ∧ (𝑎 = 𝑑 ∧ 𝑒 = 𝑓)))))

Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 399   ≠ wne 2987  ∀wral 3106  ∃wrex 3107  ∃!wreu 3108  ∅c0 4243

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-ne 2988  df-ral 3111  df-rex 3112  df-reu 3113  df-dif 3884  df-nul 4244

This theorem is referenced by: (None)