Theorem 2reu5lem3 3714

Description: Lemma for 2reu5 3715. This lemma is interesting in its own right, showing that existential restriction in the last conjunct (the "at most one" part) is optional; compare rmo2 3833. (Contributed by Alexander van der Vekens, 17-Jun-2017.)

Assertion

Ref	Expression
2reu5lem3	⊢ ((∃!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))))

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

Allowed substitution hints:   𝜑(𝑥, 𝑦)   𝐴(𝑥)   𝐵(𝑦)

Proof of Theorem 2reu5lem3

Step	Hyp	Ref	Expression
1		2reu5lem1 3712	. . 3 ⊢ (∃!𝑥 ∈ 𝐴 ∃!𝑦 ∈ 𝐵 𝜑 ↔ ∃!𝑥∃!𝑦(𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵 ∧ 𝜑))
2		2reu5lem2 3713	. . 3 ⊢ (∀𝑥 ∈ 𝐴 ∃*𝑦 ∈ 𝐵 𝜑 ↔ ∀𝑥∃*𝑦(𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵 ∧ 𝜑))
3	1, 2	anbi12i 640	. 2 ⊢ ((∃!𝑥 ∈ 𝐴 ∃!𝑦 ∈ 𝐵 𝜑 ∧ ∀𝑥 ∈ 𝐴 ∃*𝑦 ∈ 𝐵 𝜑) ↔ (∃!𝑥∃!𝑦(𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵 ∧ 𝜑) ∧ ∀𝑥∃*𝑦(𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵 ∧ 𝜑)))
4		2eu5 2680	. 2 ⊢ ((∃!𝑥∃!𝑦(𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵 ∧ 𝜑) ∧ ∀𝑥∃*𝑦(𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵 ∧ 𝜑)) ↔ (∃𝑥∃𝑦(𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵 ∧ 𝜑) ∧ ∃𝑧∃𝑤∀𝑥∀𝑦((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵 ∧ 𝜑) → (𝑥 = 𝑧 ∧ 𝑦 = 𝑤))))
5		3anass 1111	. . . . . . 7 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵 ∧ 𝜑) ↔ (𝑥 ∈ 𝐴 ∧ (𝑦 ∈ 𝐵 ∧ 𝜑)))
6	5	exbii 1880	. . . . . 6 ⊢ (∃𝑦(𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵 ∧ 𝜑) ↔ ∃𝑦(𝑥 ∈ 𝐴 ∧ (𝑦 ∈ 𝐵 ∧ 𝜑)))
7		19.42v 1985	. . . . . 6 ⊢ (∃𝑦(𝑥 ∈ 𝐴 ∧ (𝑦 ∈ 𝐵 ∧ 𝜑)) ↔ (𝑥 ∈ 𝐴 ∧ ∃𝑦(𝑦 ∈ 𝐵 ∧ 𝜑)))
8		df-rex 3087	. . . . . . . 8 ⊢ (∃𝑦 ∈ 𝐵 𝜑 ↔ ∃𝑦(𝑦 ∈ 𝐵 ∧ 𝜑))
9	8	bicomi 227	. . . . . . 7 ⊢ (∃𝑦(𝑦 ∈ 𝐵 ∧ 𝜑) ↔ ∃𝑦 ∈ 𝐵 𝜑)
10	9	anbi2i 635	. . . . . 6 ⊢ ((𝑥 ∈ 𝐴 ∧ ∃𝑦(𝑦 ∈ 𝐵 ∧ 𝜑)) ↔ (𝑥 ∈ 𝐴 ∧ ∃𝑦 ∈ 𝐵 𝜑))
11	6, 7, 10	3bitri 300	. . . . 5 ⊢ (∃𝑦(𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵 ∧ 𝜑) ↔ (𝑥 ∈ 𝐴 ∧ ∃𝑦 ∈ 𝐵 𝜑))
12	11	exbii 1880	. . . 4 ⊢ (∃𝑥∃𝑦(𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵 ∧ 𝜑) ↔ ∃𝑥(𝑥 ∈ 𝐴 ∧ ∃𝑦 ∈ 𝐵 𝜑))
13		df-rex 3087	. . . 4 ⊢ (∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝜑 ↔ ∃𝑥(𝑥 ∈ 𝐴 ∧ ∃𝑦 ∈ 𝐵 𝜑))
14	12, 13	bitr4i 281	. . 3 ⊢ (∃𝑥∃𝑦(𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵 ∧ 𝜑) ↔ ∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝜑)
15		3anan12 1112	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵 ∧ 𝜑) ↔ (𝑦 ∈ 𝐵 ∧ (𝑥 ∈ 𝐴 ∧ 𝜑)))
16	15	imbi1i 352	. . . . . . . . . 10 ⊢ (((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵 ∧ 𝜑) → (𝑥 = 𝑧 ∧ 𝑦 = 𝑤)) ↔ ((𝑦 ∈ 𝐵 ∧ (𝑥 ∈ 𝐴 ∧ 𝜑)) → (𝑥 = 𝑧 ∧ 𝑦 = 𝑤)))
17		impexp 456	. . . . . . . . . 10 ⊢ (((𝑦 ∈ 𝐵 ∧ (𝑥 ∈ 𝐴 ∧ 𝜑)) → (𝑥 = 𝑧 ∧ 𝑦 = 𝑤)) ↔ (𝑦 ∈ 𝐵 → ((𝑥 ∈ 𝐴 ∧ 𝜑) → (𝑥 = 𝑧 ∧ 𝑦 = 𝑤))))
18		impexp 456	. . . . . . . . . . 11 ⊢ (((𝑥 ∈ 𝐴 ∧ 𝜑) → (𝑥 = 𝑧 ∧ 𝑦 = 𝑤)) ↔ (𝑥 ∈ 𝐴 → (𝜑 → (𝑥 = 𝑧 ∧ 𝑦 = 𝑤))))
19	18	imbi2i 339	. . . . . . . . . 10 ⊢ ((𝑦 ∈ 𝐵 → ((𝑥 ∈ 𝐴 ∧ 𝜑) → (𝑥 = 𝑧 ∧ 𝑦 = 𝑤))) ↔ (𝑦 ∈ 𝐵 → (𝑥 ∈ 𝐴 → (𝜑 → (𝑥 = 𝑧 ∧ 𝑦 = 𝑤)))))
20	16, 17, 19	3bitri 300	. . . . . . . . 9 ⊢ (((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵 ∧ 𝜑) → (𝑥 = 𝑧 ∧ 𝑦 = 𝑤)) ↔ (𝑦 ∈ 𝐵 → (𝑥 ∈ 𝐴 → (𝜑 → (𝑥 = 𝑧 ∧ 𝑦 = 𝑤)))))
21	20	albii 1851	. . . . . . . 8 ⊢ (∀𝑦((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵 ∧ 𝜑) → (𝑥 = 𝑧 ∧ 𝑦 = 𝑤)) ↔ ∀𝑦(𝑦 ∈ 𝐵 → (𝑥 ∈ 𝐴 → (𝜑 → (𝑥 = 𝑧 ∧ 𝑦 = 𝑤)))))
22		df-ral 3077	. . . . . . . 8 ⊢ (∀𝑦 ∈ 𝐵 (𝑥 ∈ 𝐴 → (𝜑 → (𝑥 = 𝑧 ∧ 𝑦 = 𝑤))) ↔ ∀𝑦(𝑦 ∈ 𝐵 → (𝑥 ∈ 𝐴 → (𝜑 → (𝑥 = 𝑧 ∧ 𝑦 = 𝑤)))))
23		r19.21v 3187	. . . . . . . 8 ⊢ (∀𝑦 ∈ 𝐵 (𝑥 ∈ 𝐴 → (𝜑 → (𝑥 = 𝑧 ∧ 𝑦 = 𝑤))) ↔ (𝑥 ∈ 𝐴 → ∀𝑦 ∈ 𝐵 (𝜑 → (𝑥 = 𝑧 ∧ 𝑦 = 𝑤))))
24	21, 22, 23	3bitr2i 302	. . . . . . 7 ⊢ (∀𝑦((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵 ∧ 𝜑) → (𝑥 = 𝑧 ∧ 𝑦 = 𝑤)) ↔ (𝑥 ∈ 𝐴 → ∀𝑦 ∈ 𝐵 (𝜑 → (𝑥 = 𝑧 ∧ 𝑦 = 𝑤))))
25	24	albii 1851	. . . . . 6 ⊢ (∀𝑥∀𝑦((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵 ∧ 𝜑) → (𝑥 = 𝑧 ∧ 𝑦 = 𝑤)) ↔ ∀𝑥(𝑥 ∈ 𝐴 → ∀𝑦 ∈ 𝐵 (𝜑 → (𝑥 = 𝑧 ∧ 𝑦 = 𝑤))))
26		df-ral 3077	. . . . . 6 ⊢ (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 (𝜑 → (𝑥 = 𝑧 ∧ 𝑦 = 𝑤)) ↔ ∀𝑥(𝑥 ∈ 𝐴 → ∀𝑦 ∈ 𝐵 (𝜑 → (𝑥 = 𝑧 ∧ 𝑦 = 𝑤))))
27	25, 26	bitr4i 281	. . . . 5 ⊢ (∀𝑥∀𝑦((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵 ∧ 𝜑) → (𝑥 = 𝑧 ∧ 𝑦 = 𝑤)) ↔ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 (𝜑 → (𝑥 = 𝑧 ∧ 𝑦 = 𝑤)))
28	27	exbii 1880	. . . 4 ⊢ (∃𝑤∀𝑥∀𝑦((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵 ∧ 𝜑) → (𝑥 = 𝑧 ∧ 𝑦 = 𝑤)) ↔ ∃𝑤∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 (𝜑 → (𝑥 = 𝑧 ∧ 𝑦 = 𝑤)))
29	28	exbii 1880	. . 3 ⊢ (∃𝑧∃𝑤∀𝑥∀𝑦((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵 ∧ 𝜑) → (𝑥 = 𝑧 ∧ 𝑦 = 𝑤)) ↔ ∃𝑧∃𝑤∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 (𝜑 → (𝑥 = 𝑧 ∧ 𝑦 = 𝑤)))
30	14, 29	anbi12i 640	. 2 ⊢ ((∃𝑥∃𝑦(𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵 ∧ 𝜑) ∧ ∃𝑧∃𝑤∀𝑥∀𝑦((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵 ∧ 𝜑) → (𝑥 = 𝑧 ∧ 𝑦 = 𝑤))) ↔ (∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝜑 ∧ ∃𝑧∃𝑤∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 (𝜑 → (𝑥 = 𝑧 ∧ 𝑦 = 𝑤))))
31	3, 4, 30	3bitri 300	1 ⊢ ((∃!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))))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   ∧ w3a 1103  ∀wal 1567  ∃wex 1811   ∈ wcel 2144  ∃*wmo 2562  ∃!weu 2593  ∀wral 3076  ∃wrex 3086  ∃!wreu 3363  ∃*wrmo 3364

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-10 2177  ax-11 2193  ax-12 2212

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1572  df-ex 1812  df-nf 1816  df-mo 2564  df-eu 2594  df-ral 3077  df-rex 3087  df-rmo 3365  df-reu 3366

This theorem is used by:  2reu5  3715