Theorem kmlem15 9851

Description: Lemma for 5-quantifier AC of Kurt Maes, Th. 4, part of 5 <=> 4. (Contributed by NM, 4-Apr-2004.)

Hypotheses

Ref	Expression
kmlem14.1	⊢ (𝜑 ↔ (𝑧 ∈ 𝑦 → ((𝑣 ∈ 𝑥 ∧ 𝑦 ≠ 𝑣) ∧ 𝑧 ∈ 𝑣)))
kmlem14.2	⊢ (𝜓 ↔ (𝑧 ∈ 𝑥 → ((𝑣 ∈ 𝑧 ∧ 𝑣 ∈ 𝑦) ∧ ((𝑢 ∈ 𝑧 ∧ 𝑢 ∈ 𝑦) → 𝑢 = 𝑣))))
kmlem14.3	⊢ (𝜒 ↔ ∀𝑧 ∈ 𝑥 ∃!𝑣 𝑣 ∈ (𝑧 ∩ 𝑦))

Assertion

Ref	Expression
kmlem15	⊢ ((¬ 𝑦 ∈ 𝑥 ∧ 𝜒) ↔ ∀𝑧∃𝑣∀𝑢(¬ 𝑦 ∈ 𝑥 ∧ 𝜓))

Distinct variable groups:   𝑥,𝑦,𝑧,𝑣,𝑢   𝜑,𝑢

Allowed substitution hints:   𝜑(𝑥,𝑦,𝑧,𝑣)   𝜓(𝑥,𝑦,𝑧,𝑣,𝑢)   𝜒(𝑥,𝑦,𝑧,𝑣,𝑢)

Proof of Theorem kmlem15

Step	Hyp	Ref	Expression
1		kmlem14.3	. . . 4 ⊢ (𝜒 ↔ ∀𝑧 ∈ 𝑥 ∃!𝑣 𝑣 ∈ (𝑧 ∩ 𝑦))
2		nfv 1918	. . . . . . 7 ⊢ Ⅎ𝑢 𝑣 ∈ (𝑧 ∩ 𝑦)
3	2	eu1 2612	. . . . . 6 ⊢ (∃!𝑣 𝑣 ∈ (𝑧 ∩ 𝑦) ↔ ∃𝑣(𝑣 ∈ (𝑧 ∩ 𝑦) ∧ ∀𝑢([𝑢 / 𝑣]𝑣 ∈ (𝑧 ∩ 𝑦) → 𝑣 = 𝑢)))
4		elin 3899	. . . . . . . . 9 ⊢ (𝑣 ∈ (𝑧 ∩ 𝑦) ↔ (𝑣 ∈ 𝑧 ∧ 𝑣 ∈ 𝑦))
5		clelsb1 2866	. . . . . . . . . . . 12 ⊢ ([𝑢 / 𝑣]𝑣 ∈ (𝑧 ∩ 𝑦) ↔ 𝑢 ∈ (𝑧 ∩ 𝑦))
6		elin 3899	. . . . . . . . . . . 12 ⊢ (𝑢 ∈ (𝑧 ∩ 𝑦) ↔ (𝑢 ∈ 𝑧 ∧ 𝑢 ∈ 𝑦))
7	5, 6	bitri 274	. . . . . . . . . . 11 ⊢ ([𝑢 / 𝑣]𝑣 ∈ (𝑧 ∩ 𝑦) ↔ (𝑢 ∈ 𝑧 ∧ 𝑢 ∈ 𝑦))
8		equcom 2022	. . . . . . . . . . 11 ⊢ (𝑣 = 𝑢 ↔ 𝑢 = 𝑣)
9	7, 8	imbi12i 350	. . . . . . . . . 10 ⊢ (([𝑢 / 𝑣]𝑣 ∈ (𝑧 ∩ 𝑦) → 𝑣 = 𝑢) ↔ ((𝑢 ∈ 𝑧 ∧ 𝑢 ∈ 𝑦) → 𝑢 = 𝑣))
10	9	albii 1823	. . . . . . . . 9 ⊢ (∀𝑢([𝑢 / 𝑣]𝑣 ∈ (𝑧 ∩ 𝑦) → 𝑣 = 𝑢) ↔ ∀𝑢((𝑢 ∈ 𝑧 ∧ 𝑢 ∈ 𝑦) → 𝑢 = 𝑣))
11	4, 10	anbi12i 626	. . . . . . . 8 ⊢ ((𝑣 ∈ (𝑧 ∩ 𝑦) ∧ ∀𝑢([𝑢 / 𝑣]𝑣 ∈ (𝑧 ∩ 𝑦) → 𝑣 = 𝑢)) ↔ ((𝑣 ∈ 𝑧 ∧ 𝑣 ∈ 𝑦) ∧ ∀𝑢((𝑢 ∈ 𝑧 ∧ 𝑢 ∈ 𝑦) → 𝑢 = 𝑣)))
12		19.28v 1995	. . . . . . . 8 ⊢ (∀𝑢((𝑣 ∈ 𝑧 ∧ 𝑣 ∈ 𝑦) ∧ ((𝑢 ∈ 𝑧 ∧ 𝑢 ∈ 𝑦) → 𝑢 = 𝑣)) ↔ ((𝑣 ∈ 𝑧 ∧ 𝑣 ∈ 𝑦) ∧ ∀𝑢((𝑢 ∈ 𝑧 ∧ 𝑢 ∈ 𝑦) → 𝑢 = 𝑣)))
13	11, 12	bitr4i 277	. . . . . . 7 ⊢ ((𝑣 ∈ (𝑧 ∩ 𝑦) ∧ ∀𝑢([𝑢 / 𝑣]𝑣 ∈ (𝑧 ∩ 𝑦) → 𝑣 = 𝑢)) ↔ ∀𝑢((𝑣 ∈ 𝑧 ∧ 𝑣 ∈ 𝑦) ∧ ((𝑢 ∈ 𝑧 ∧ 𝑢 ∈ 𝑦) → 𝑢 = 𝑣)))
14	13	exbii 1851	. . . . . 6 ⊢ (∃𝑣(𝑣 ∈ (𝑧 ∩ 𝑦) ∧ ∀𝑢([𝑢 / 𝑣]𝑣 ∈ (𝑧 ∩ 𝑦) → 𝑣 = 𝑢)) ↔ ∃𝑣∀𝑢((𝑣 ∈ 𝑧 ∧ 𝑣 ∈ 𝑦) ∧ ((𝑢 ∈ 𝑧 ∧ 𝑢 ∈ 𝑦) → 𝑢 = 𝑣)))
15	3, 14	bitri 274	. . . . 5 ⊢ (∃!𝑣 𝑣 ∈ (𝑧 ∩ 𝑦) ↔ ∃𝑣∀𝑢((𝑣 ∈ 𝑧 ∧ 𝑣 ∈ 𝑦) ∧ ((𝑢 ∈ 𝑧 ∧ 𝑢 ∈ 𝑦) → 𝑢 = 𝑣)))
16	15	ralbii 3090	. . . 4 ⊢ (∀𝑧 ∈ 𝑥 ∃!𝑣 𝑣 ∈ (𝑧 ∩ 𝑦) ↔ ∀𝑧 ∈ 𝑥 ∃𝑣∀𝑢((𝑣 ∈ 𝑧 ∧ 𝑣 ∈ 𝑦) ∧ ((𝑢 ∈ 𝑧 ∧ 𝑢 ∈ 𝑦) → 𝑢 = 𝑣)))
17		df-ral 3068	. . . . 5 ⊢ (∀𝑧 ∈ 𝑥 ∃𝑣∀𝑢((𝑣 ∈ 𝑧 ∧ 𝑣 ∈ 𝑦) ∧ ((𝑢 ∈ 𝑧 ∧ 𝑢 ∈ 𝑦) → 𝑢 = 𝑣)) ↔ ∀𝑧(𝑧 ∈ 𝑥 → ∃𝑣∀𝑢((𝑣 ∈ 𝑧 ∧ 𝑣 ∈ 𝑦) ∧ ((𝑢 ∈ 𝑧 ∧ 𝑢 ∈ 𝑦) → 𝑢 = 𝑣))))
18		kmlem14.2	. . . . . . . . . 10 ⊢ (𝜓 ↔ (𝑧 ∈ 𝑥 → ((𝑣 ∈ 𝑧 ∧ 𝑣 ∈ 𝑦) ∧ ((𝑢 ∈ 𝑧 ∧ 𝑢 ∈ 𝑦) → 𝑢 = 𝑣))))
19	18	albii 1823	. . . . . . . . 9 ⊢ (∀𝑢𝜓 ↔ ∀𝑢(𝑧 ∈ 𝑥 → ((𝑣 ∈ 𝑧 ∧ 𝑣 ∈ 𝑦) ∧ ((𝑢 ∈ 𝑧 ∧ 𝑢 ∈ 𝑦) → 𝑢 = 𝑣))))
20		19.21v 1943	. . . . . . . . 9 ⊢ (∀𝑢(𝑧 ∈ 𝑥 → ((𝑣 ∈ 𝑧 ∧ 𝑣 ∈ 𝑦) ∧ ((𝑢 ∈ 𝑧 ∧ 𝑢 ∈ 𝑦) → 𝑢 = 𝑣))) ↔ (𝑧 ∈ 𝑥 → ∀𝑢((𝑣 ∈ 𝑧 ∧ 𝑣 ∈ 𝑦) ∧ ((𝑢 ∈ 𝑧 ∧ 𝑢 ∈ 𝑦) → 𝑢 = 𝑣))))
21	19, 20	bitri 274	. . . . . . . 8 ⊢ (∀𝑢𝜓 ↔ (𝑧 ∈ 𝑥 → ∀𝑢((𝑣 ∈ 𝑧 ∧ 𝑣 ∈ 𝑦) ∧ ((𝑢 ∈ 𝑧 ∧ 𝑢 ∈ 𝑦) → 𝑢 = 𝑣))))
22	21	exbii 1851	. . . . . . 7 ⊢ (∃𝑣∀𝑢𝜓 ↔ ∃𝑣(𝑧 ∈ 𝑥 → ∀𝑢((𝑣 ∈ 𝑧 ∧ 𝑣 ∈ 𝑦) ∧ ((𝑢 ∈ 𝑧 ∧ 𝑢 ∈ 𝑦) → 𝑢 = 𝑣))))
23		19.37v 1996	. . . . . . 7 ⊢ (∃𝑣(𝑧 ∈ 𝑥 → ∀𝑢((𝑣 ∈ 𝑧 ∧ 𝑣 ∈ 𝑦) ∧ ((𝑢 ∈ 𝑧 ∧ 𝑢 ∈ 𝑦) → 𝑢 = 𝑣))) ↔ (𝑧 ∈ 𝑥 → ∃𝑣∀𝑢((𝑣 ∈ 𝑧 ∧ 𝑣 ∈ 𝑦) ∧ ((𝑢 ∈ 𝑧 ∧ 𝑢 ∈ 𝑦) → 𝑢 = 𝑣))))
24	22, 23	bitri 274	. . . . . 6 ⊢ (∃𝑣∀𝑢𝜓 ↔ (𝑧 ∈ 𝑥 → ∃𝑣∀𝑢((𝑣 ∈ 𝑧 ∧ 𝑣 ∈ 𝑦) ∧ ((𝑢 ∈ 𝑧 ∧ 𝑢 ∈ 𝑦) → 𝑢 = 𝑣))))
25	24	albii 1823	. . . . 5 ⊢ (∀𝑧∃𝑣∀𝑢𝜓 ↔ ∀𝑧(𝑧 ∈ 𝑥 → ∃𝑣∀𝑢((𝑣 ∈ 𝑧 ∧ 𝑣 ∈ 𝑦) ∧ ((𝑢 ∈ 𝑧 ∧ 𝑢 ∈ 𝑦) → 𝑢 = 𝑣))))
26	17, 25	bitr4i 277	. . . 4 ⊢ (∀𝑧 ∈ 𝑥 ∃𝑣∀𝑢((𝑣 ∈ 𝑧 ∧ 𝑣 ∈ 𝑦) ∧ ((𝑢 ∈ 𝑧 ∧ 𝑢 ∈ 𝑦) → 𝑢 = 𝑣)) ↔ ∀𝑧∃𝑣∀𝑢𝜓)
27	1, 16, 26	3bitri 296	. . 3 ⊢ (𝜒 ↔ ∀𝑧∃𝑣∀𝑢𝜓)
28	27	anbi2i 622	. 2 ⊢ ((¬ 𝑦 ∈ 𝑥 ∧ 𝜒) ↔ (¬ 𝑦 ∈ 𝑥 ∧ ∀𝑧∃𝑣∀𝑢𝜓))
29		19.28v 1995	. 2 ⊢ (∀𝑧(¬ 𝑦 ∈ 𝑥 ∧ ∃𝑣∀𝑢𝜓) ↔ (¬ 𝑦 ∈ 𝑥 ∧ ∀𝑧∃𝑣∀𝑢𝜓))
30		19.28v 1995	. . . . 5 ⊢ (∀𝑢(¬ 𝑦 ∈ 𝑥 ∧ 𝜓) ↔ (¬ 𝑦 ∈ 𝑥 ∧ ∀𝑢𝜓))
31	30	exbii 1851	. . . 4 ⊢ (∃𝑣∀𝑢(¬ 𝑦 ∈ 𝑥 ∧ 𝜓) ↔ ∃𝑣(¬ 𝑦 ∈ 𝑥 ∧ ∀𝑢𝜓))
32		19.42v 1958	. . . 4 ⊢ (∃𝑣(¬ 𝑦 ∈ 𝑥 ∧ ∀𝑢𝜓) ↔ (¬ 𝑦 ∈ 𝑥 ∧ ∃𝑣∀𝑢𝜓))
33	31, 32	bitr2i 275	. . 3 ⊢ ((¬ 𝑦 ∈ 𝑥 ∧ ∃𝑣∀𝑢𝜓) ↔ ∃𝑣∀𝑢(¬ 𝑦 ∈ 𝑥 ∧ 𝜓))
34	33	albii 1823	. 2 ⊢ (∀𝑧(¬ 𝑦 ∈ 𝑥 ∧ ∃𝑣∀𝑢𝜓) ↔ ∀𝑧∃𝑣∀𝑢(¬ 𝑦 ∈ 𝑥 ∧ 𝜓))
35	28, 29, 34	3bitr2i 298	1 ⊢ ((¬ 𝑦 ∈ 𝑥 ∧ 𝜒) ↔ ∀𝑧∃𝑣∀𝑢(¬ 𝑦 ∈ 𝑥 ∧ 𝜓))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 205   ∧ wa 395  ∀wal 1537  ∃wex 1783  [wsb 2068   ∈ wcel 2108  ∃!weu 2568   ≠ wne 2942  ∀wral 3063   ∩ cin 3882

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2156  ax-12 2173  ax-ext 2709

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-tru 1542  df-ex 1784  df-nf 1788  df-sb 2069  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2817  df-ral 3068  df-v 3424  df-in 3890

This theorem is referenced by:  kmlem16  9852