Theorem 2reu5 3722

Description: Double restricted existential uniqueness in terms of restricted existential quantification and restricted universal quantification, analogous to 2eu5 2683 and reu3 3691. (Contributed by Alexander van der Vekens, 17-Jun-2017.)

Assertion

Ref	Expression
2reu5	⊢ ((∃!𝑥 ∈ 𝐴 ∃!𝑦 ∈ 𝐵 𝜑 ∧ ∀𝑥 ∈ 𝐴 ∃*𝑦 ∈ 𝐵 𝜑) ↔ (∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝜑 ∧ ∃𝑧 ∈ 𝐴 ∃𝑤 ∈ 𝐵 ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 (𝜑 → (𝑥 = 𝑧 ∧ 𝑦 = 𝑤))))

Distinct variable groups:   𝑦,𝑤,𝑧,𝐴,𝑥   𝑤,𝐵   𝑥,𝑧,𝐵,𝑦   𝜑,𝑤,𝑧   𝑥,𝐴   𝑦,𝐵

Allowed substitution hints:   𝜑(𝑥,𝑦)

Proof of Theorem 2reu5

Step	Hyp	Ref	Expression
1		r19.29r 3127	. . . . . . . 8 ⊢ ((∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝜑 ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 (𝜑 → (𝑥 = 𝑧 ∧ 𝑦 = 𝑤))) → ∃𝑥 ∈ 𝐴 (∃𝑦 ∈ 𝐵 𝜑 ∧ ∀𝑦 ∈ 𝐵 (𝜑 → (𝑥 = 𝑧 ∧ 𝑦 = 𝑤))))
2		r19.29r 3127	. . . . . . . . 9 ⊢ ((∃𝑦 ∈ 𝐵 𝜑 ∧ ∀𝑦 ∈ 𝐵 (𝜑 → (𝑥 = 𝑧 ∧ 𝑦 = 𝑤))) → ∃𝑦 ∈ 𝐵 (𝜑 ∧ (𝜑 → (𝑥 = 𝑧 ∧ 𝑦 = 𝑤))))
3	2	reximi 3101	. . . . . . . 8 ⊢ (∃𝑥 ∈ 𝐴 (∃𝑦 ∈ 𝐵 𝜑 ∧ ∀𝑦 ∈ 𝐵 (𝜑 → (𝑥 = 𝑧 ∧ 𝑦 = 𝑤))) → ∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 (𝜑 ∧ (𝜑 → (𝑥 = 𝑧 ∧ 𝑦 = 𝑤))))
4		pm3.35 812	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝜑 → (𝑥 = 𝑧 ∧ 𝑦 = 𝑤))) → (𝑥 = 𝑧 ∧ 𝑦 = 𝑤))
5	4	reximi 3101	. . . . . . . . 9 ⊢ (∃𝑦 ∈ 𝐵 (𝜑 ∧ (𝜑 → (𝑥 = 𝑧 ∧ 𝑦 = 𝑤))) → ∃𝑦 ∈ 𝐵 (𝑥 = 𝑧 ∧ 𝑦 = 𝑤))
6	5	reximi 3101	. . . . . . . 8 ⊢ (∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 (𝜑 ∧ (𝜑 → (𝑥 = 𝑧 ∧ 𝑦 = 𝑤))) → ∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 (𝑥 = 𝑧 ∧ 𝑦 = 𝑤))
7		eleq1w 2846	. . . . . . . . . . . . 13 ⊢ (𝑥 = 𝑧 → (𝑥 ∈ 𝐴 ↔ 𝑧 ∈ 𝐴))
8		eleq1w 2846	. . . . . . . . . . . . 13 ⊢ (𝑦 = 𝑤 → (𝑦 ∈ 𝐵 ↔ 𝑤 ∈ 𝐵))
9	7, 8	bi2anan9 647	. . . . . . . . . . . 12 ⊢ ((𝑥 = 𝑧 ∧ 𝑦 = 𝑤) → ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ↔ (𝑧 ∈ 𝐴 ∧ 𝑤 ∈ 𝐵)))
10	9	biimpac 482	. . . . . . . . . . 11 ⊢ (((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ (𝑥 = 𝑧 ∧ 𝑦 = 𝑤)) → (𝑧 ∈ 𝐴 ∧ 𝑤 ∈ 𝐵))
11	10	ancomd 465	. . . . . . . . . 10 ⊢ (((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ (𝑥 = 𝑧 ∧ 𝑦 = 𝑤)) → (𝑤 ∈ 𝐵 ∧ 𝑧 ∈ 𝐴))
12	11	ex 416	. . . . . . . . 9 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) → ((𝑥 = 𝑧 ∧ 𝑦 = 𝑤) → (𝑤 ∈ 𝐵 ∧ 𝑧 ∈ 𝐴)))
13	12	rexlimivv 3205	. . . . . . . 8 ⊢ (∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 (𝑥 = 𝑧 ∧ 𝑦 = 𝑤) → (𝑤 ∈ 𝐵 ∧ 𝑧 ∈ 𝐴))
14	1, 3, 6, 13	4syl 19	. . . . . . 7 ⊢ ((∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝜑 ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 (𝜑 → (𝑥 = 𝑧 ∧ 𝑦 = 𝑤))) → (𝑤 ∈ 𝐵 ∧ 𝑧 ∈ 𝐴))
15	14	ex 416	. . . . . 6 ⊢ (∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝜑 → (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 (𝜑 → (𝑥 = 𝑧 ∧ 𝑦 = 𝑤)) → (𝑤 ∈ 𝐵 ∧ 𝑧 ∈ 𝐴)))
16	15	pm4.71rd 570	. . . . 5 ⊢ (∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝜑 → (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 (𝜑 → (𝑥 = 𝑧 ∧ 𝑦 = 𝑤)) ↔ ((𝑤 ∈ 𝐵 ∧ 𝑧 ∈ 𝐴) ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 (𝜑 → (𝑥 = 𝑧 ∧ 𝑦 = 𝑤)))))
17		anass 472	. . . . 5 ⊢ (((𝑤 ∈ 𝐵 ∧ 𝑧 ∈ 𝐴) ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 (𝜑 → (𝑥 = 𝑧 ∧ 𝑦 = 𝑤))) ↔ (𝑤 ∈ 𝐵 ∧ (𝑧 ∈ 𝐴 ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 (𝜑 → (𝑥 = 𝑧 ∧ 𝑦 = 𝑤)))))
18	16, 17	bitrdi 289	. . . 4 ⊢ (∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝜑 → (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 (𝜑 → (𝑥 = 𝑧 ∧ 𝑦 = 𝑤)) ↔ (𝑤 ∈ 𝐵 ∧ (𝑧 ∈ 𝐴 ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 (𝜑 → (𝑥 = 𝑧 ∧ 𝑦 = 𝑤))))))
19	18	2exbidv 1945	. . 3 ⊢ (∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝜑 → (∃𝑧∃𝑤∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 (𝜑 → (𝑥 = 𝑧 ∧ 𝑦 = 𝑤)) ↔ ∃𝑧∃𝑤(𝑤 ∈ 𝐵 ∧ (𝑧 ∈ 𝐴 ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 (𝜑 → (𝑥 = 𝑧 ∧ 𝑦 = 𝑤))))))
20	19	pm5.32i 582	. 2 ⊢ ((∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝜑 ∧ ∃𝑧∃𝑤∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 (𝜑 → (𝑥 = 𝑧 ∧ 𝑦 = 𝑤))) ↔ (∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝜑 ∧ ∃𝑧∃𝑤(𝑤 ∈ 𝐵 ∧ (𝑧 ∈ 𝐴 ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 (𝜑 → (𝑥 = 𝑧 ∧ 𝑦 = 𝑤))))))
21		2reu5lem3 3721	. 2 ⊢ ((∃!𝑥 ∈ 𝐴 ∃!𝑦 ∈ 𝐵 𝜑 ∧ ∀𝑥 ∈ 𝐴 ∃*𝑦 ∈ 𝐵 𝜑) ↔ (∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝜑 ∧ ∃𝑧∃𝑤∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 (𝜑 → (𝑥 = 𝑧 ∧ 𝑦 = 𝑤))))
22		df-rex 3088	. . . 4 ⊢ (∃𝑧 ∈ 𝐴 ∃𝑤 ∈ 𝐵 ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 (𝜑 → (𝑥 = 𝑧 ∧ 𝑦 = 𝑤)) ↔ ∃𝑧(𝑧 ∈ 𝐴 ∧ ∃𝑤 ∈ 𝐵 ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 (𝜑 → (𝑥 = 𝑧 ∧ 𝑦 = 𝑤))))
23		r19.42v 3195	. . . . . 6 ⊢ (∃𝑤 ∈ 𝐵 (𝑧 ∈ 𝐴 ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 (𝜑 → (𝑥 = 𝑧 ∧ 𝑦 = 𝑤))) ↔ (𝑧 ∈ 𝐴 ∧ ∃𝑤 ∈ 𝐵 ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 (𝜑 → (𝑥 = 𝑧 ∧ 𝑦 = 𝑤))))
24		df-rex 3088	. . . . . 6 ⊢ (∃𝑤 ∈ 𝐵 (𝑧 ∈ 𝐴 ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 (𝜑 → (𝑥 = 𝑧 ∧ 𝑦 = 𝑤))) ↔ ∃𝑤(𝑤 ∈ 𝐵 ∧ (𝑧 ∈ 𝐴 ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 (𝜑 → (𝑥 = 𝑧 ∧ 𝑦 = 𝑤)))))
25	23, 24	bitr3i 279	. . . . 5 ⊢ ((𝑧 ∈ 𝐴 ∧ ∃𝑤 ∈ 𝐵 ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 (𝜑 → (𝑥 = 𝑧 ∧ 𝑦 = 𝑤))) ↔ ∃𝑤(𝑤 ∈ 𝐵 ∧ (𝑧 ∈ 𝐴 ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 (𝜑 → (𝑥 = 𝑧 ∧ 𝑦 = 𝑤)))))
26	25	exbii 1869	. . . 4 ⊢ (∃𝑧(𝑧 ∈ 𝐴 ∧ ∃𝑤 ∈ 𝐵 ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 (𝜑 → (𝑥 = 𝑧 ∧ 𝑦 = 𝑤))) ↔ ∃𝑧∃𝑤(𝑤 ∈ 𝐵 ∧ (𝑧 ∈ 𝐴 ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 (𝜑 → (𝑥 = 𝑧 ∧ 𝑦 = 𝑤)))))
27	22, 26	bitri 277	. . 3 ⊢ (∃𝑧 ∈ 𝐴 ∃𝑤 ∈ 𝐵 ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 (𝜑 → (𝑥 = 𝑧 ∧ 𝑦 = 𝑤)) ↔ ∃𝑧∃𝑤(𝑤 ∈ 𝐵 ∧ (𝑧 ∈ 𝐴 ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 (𝜑 → (𝑥 = 𝑧 ∧ 𝑦 = 𝑤)))))
28	27	anbi2i 632	. 2 ⊢ ((∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝜑 ∧ ∃𝑧 ∈ 𝐴 ∃𝑤 ∈ 𝐵 ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 (𝜑 → (𝑥 = 𝑧 ∧ 𝑦 = 𝑤))) ↔ (∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝜑 ∧ ∃𝑧∃𝑤(𝑤 ∈ 𝐵 ∧ (𝑧 ∈ 𝐴 ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 (𝜑 → (𝑥 = 𝑧 ∧ 𝑦 = 𝑤))))))
29	20, 21, 28	3bitr4i 305	1 ⊢ ((∃!𝑥 ∈ 𝐴 ∃!𝑦 ∈ 𝐵 𝜑 ∧ ∀𝑥 ∈ 𝐴 ∃*𝑦 ∈ 𝐵 𝜑) ↔ (∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝜑 ∧ ∃𝑧 ∈ 𝐴 ∃𝑤 ∈ 𝐵 ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 (𝜑 → (𝑥 = 𝑧 ∧ 𝑦 = 𝑤))))

Syntax hints:   → wi 4   ↔ wb 208   ∧ wa 399  ∃wex 1800   ∈ wcel 2143  ∀wral 3077  ∃wrex 3087  ∃!wreu 3366  ∃*wrmo 3367

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1816  ax-4 1830  ax-5 1931  ax-6 1988  ax-7 2029  ax-8 2145  ax-10 2176  ax-11 2192  ax-12 2213

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3an 1101  df-tru 1564  df-ex 1801  df-nf 1805  df-mo 2567  df-eu 2597  df-clel 2838  df-ral 3078  df-rex 3088  df-rmo 3368  df-reu 3369

This theorem is referenced by: (None)