Theorem bnj1174 35499

Description: Technical lemma for bnj69 35506. This lemma may no longer be used or have become an indirect lemma of the theorem in question (i.e. a lemma of a lemma... of the theorem). (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)

Hypotheses

Ref	Expression
bnj1174.3	⊢ 𝐶 = ( trCl(𝑋, 𝐴, 𝑅) ∩ 𝐵)
bnj1174.59	⊢ ∃𝑧∀𝑤((𝜑 ∧ 𝜓) → (𝑧 ∈ 𝐶 ∧ (𝜃 → (𝑤𝑅𝑧 → ¬ 𝑤 ∈ 𝐶))))
bnj1174.74	⊢ (𝜃 → (𝑤𝑅𝑧 → 𝑤 ∈ trCl(𝑋, 𝐴, 𝑅)))

Assertion

Ref	Expression
bnj1174	⊢ ∃𝑧∀𝑤((𝜑 ∧ 𝜓) → ((𝜑 ∧ 𝜓 ∧ 𝑧 ∈ 𝐶) ∧ (𝜃 → (𝑤𝑅𝑧 → ¬ 𝑤 ∈ 𝐵))))

Proof of Theorem bnj1174

Step	Hyp	Ref	Expression
1		bnj1174.59	. . . . 5 ⊢ ∃𝑧∀𝑤((𝜑 ∧ 𝜓) → (𝑧 ∈ 𝐶 ∧ (𝜃 → (𝑤𝑅𝑧 → ¬ 𝑤 ∈ 𝐶))))
2		bnj1174.74	. . . . . . . . . . . 12 ⊢ (𝜃 → (𝑤𝑅𝑧 → 𝑤 ∈ trCl(𝑋, 𝐴, 𝑅)))
3		bnj1174.3	. . . . . . . . . . . . . . . 16 ⊢ 𝐶 = ( trCl(𝑋, 𝐴, 𝑅) ∩ 𝐵)
4	3	eleq2i 2854	. . . . . . . . . . . . . . 15 ⊢ (𝑤 ∈ 𝐶 ↔ 𝑤 ∈ ( trCl(𝑋, 𝐴, 𝑅) ∩ 𝐵))
5	4	notbii 323	. . . . . . . . . . . . . 14 ⊢ (¬ 𝑤 ∈ 𝐶 ↔ ¬ 𝑤 ∈ ( trCl(𝑋, 𝐴, 𝑅) ∩ 𝐵))
6		ianor 997	. . . . . . . . . . . . . . . . 17 ⊢ (¬ (𝑤 ∈ trCl(𝑋, 𝐴, 𝑅) ∧ 𝑤 ∈ 𝐵) ↔ (¬ 𝑤 ∈ trCl(𝑋, 𝐴, 𝑅) ∨ ¬ 𝑤 ∈ 𝐵))
7		elin 3918	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑤 ∈ ( trCl(𝑋, 𝐴, 𝑅) ∩ 𝐵) ↔ (𝑤 ∈ trCl(𝑋, 𝐴, 𝑅) ∧ 𝑤 ∈ 𝐵))
8	7	notbii 323	. . . . . . . . . . . . . . . . 17 ⊢ (¬ 𝑤 ∈ ( trCl(𝑋, 𝐴, 𝑅) ∩ 𝐵) ↔ ¬ (𝑤 ∈ trCl(𝑋, 𝐴, 𝑅) ∧ 𝑤 ∈ 𝐵))
9		pm4.62 870	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑤 ∈ trCl(𝑋, 𝐴, 𝑅) → ¬ 𝑤 ∈ 𝐵) ↔ (¬ 𝑤 ∈ trCl(𝑋, 𝐴, 𝑅) ∨ ¬ 𝑤 ∈ 𝐵))
10	6, 8, 9	3bitr4i 306	. . . . . . . . . . . . . . . 16 ⊢ (¬ 𝑤 ∈ ( trCl(𝑋, 𝐴, 𝑅) ∩ 𝐵) ↔ (𝑤 ∈ trCl(𝑋, 𝐴, 𝑅) → ¬ 𝑤 ∈ 𝐵))
11	10	biimpi 219	. . . . . . . . . . . . . . 15 ⊢ (¬ 𝑤 ∈ ( trCl(𝑋, 𝐴, 𝑅) ∩ 𝐵) → (𝑤 ∈ trCl(𝑋, 𝐴, 𝑅) → ¬ 𝑤 ∈ 𝐵))
12	11	impcom 413	. . . . . . . . . . . . . 14 ⊢ ((𝑤 ∈ trCl(𝑋, 𝐴, 𝑅) ∧ ¬ 𝑤 ∈ ( trCl(𝑋, 𝐴, 𝑅) ∩ 𝐵)) → ¬ 𝑤 ∈ 𝐵)
13	5, 12	sylan2b 606	. . . . . . . . . . . . 13 ⊢ ((𝑤 ∈ trCl(𝑋, 𝐴, 𝑅) ∧ ¬ 𝑤 ∈ 𝐶) → ¬ 𝑤 ∈ 𝐵)
14	13	ex 418	. . . . . . . . . . . 12 ⊢ (𝑤 ∈ trCl(𝑋, 𝐴, 𝑅) → (¬ 𝑤 ∈ 𝐶 → ¬ 𝑤 ∈ 𝐵))
15	2, 14	syl6 36	. . . . . . . . . . 11 ⊢ (𝜃 → (𝑤𝑅𝑧 → (¬ 𝑤 ∈ 𝐶 → ¬ 𝑤 ∈ 𝐵)))
16	15	a2d 30	. . . . . . . . . 10 ⊢ (𝜃 → ((𝑤𝑅𝑧 → ¬ 𝑤 ∈ 𝐶) → (𝑤𝑅𝑧 → ¬ 𝑤 ∈ 𝐵)))
17	16	biantru 539	. . . . . . . . 9 ⊢ ((𝑧 ∈ 𝐶 ∧ (𝜃 → (𝑤𝑅𝑧 → ¬ 𝑤 ∈ 𝐶))) ↔ ((𝑧 ∈ 𝐶 ∧ (𝜃 → (𝑤𝑅𝑧 → ¬ 𝑤 ∈ 𝐶))) ∧ (𝜃 → ((𝑤𝑅𝑧 → ¬ 𝑤 ∈ 𝐶) → (𝑤𝑅𝑧 → ¬ 𝑤 ∈ 𝐵)))))
18		df-3an 1105	. . . . . . . . 9 ⊢ ((𝑧 ∈ 𝐶 ∧ (𝜃 → (𝑤𝑅𝑧 → ¬ 𝑤 ∈ 𝐶)) ∧ (𝜃 → ((𝑤𝑅𝑧 → ¬ 𝑤 ∈ 𝐶) → (𝑤𝑅𝑧 → ¬ 𝑤 ∈ 𝐵)))) ↔ ((𝑧 ∈ 𝐶 ∧ (𝜃 → (𝑤𝑅𝑧 → ¬ 𝑤 ∈ 𝐶))) ∧ (𝜃 → ((𝑤𝑅𝑧 → ¬ 𝑤 ∈ 𝐶) → (𝑤𝑅𝑧 → ¬ 𝑤 ∈ 𝐵)))))
19		3anass 1111	. . . . . . . . 9 ⊢ ((𝑧 ∈ 𝐶 ∧ (𝜃 → (𝑤𝑅𝑧 → ¬ 𝑤 ∈ 𝐶)) ∧ (𝜃 → ((𝑤𝑅𝑧 → ¬ 𝑤 ∈ 𝐶) → (𝑤𝑅𝑧 → ¬ 𝑤 ∈ 𝐵)))) ↔ (𝑧 ∈ 𝐶 ∧ ((𝜃 → (𝑤𝑅𝑧 → ¬ 𝑤 ∈ 𝐶)) ∧ (𝜃 → ((𝑤𝑅𝑧 → ¬ 𝑤 ∈ 𝐶) → (𝑤𝑅𝑧 → ¬ 𝑤 ∈ 𝐵))))))
20	17, 18, 19	3bitr2i 302	. . . . . . . 8 ⊢ ((𝑧 ∈ 𝐶 ∧ (𝜃 → (𝑤𝑅𝑧 → ¬ 𝑤 ∈ 𝐶))) ↔ (𝑧 ∈ 𝐶 ∧ ((𝜃 → (𝑤𝑅𝑧 → ¬ 𝑤 ∈ 𝐶)) ∧ (𝜃 → ((𝑤𝑅𝑧 → ¬ 𝑤 ∈ 𝐶) → (𝑤𝑅𝑧 → ¬ 𝑤 ∈ 𝐵))))))
21	20	imbi2i 339	. . . . . . 7 ⊢ (((𝜑 ∧ 𝜓) → (𝑧 ∈ 𝐶 ∧ (𝜃 → (𝑤𝑅𝑧 → ¬ 𝑤 ∈ 𝐶)))) ↔ ((𝜑 ∧ 𝜓) → (𝑧 ∈ 𝐶 ∧ ((𝜃 → (𝑤𝑅𝑧 → ¬ 𝑤 ∈ 𝐶)) ∧ (𝜃 → ((𝑤𝑅𝑧 → ¬ 𝑤 ∈ 𝐶) → (𝑤𝑅𝑧 → ¬ 𝑤 ∈ 𝐵)))))))
22	21	albii 1852	. . . . . 6 ⊢ (∀𝑤((𝜑 ∧ 𝜓) → (𝑧 ∈ 𝐶 ∧ (𝜃 → (𝑤𝑅𝑧 → ¬ 𝑤 ∈ 𝐶)))) ↔ ∀𝑤((𝜑 ∧ 𝜓) → (𝑧 ∈ 𝐶 ∧ ((𝜃 → (𝑤𝑅𝑧 → ¬ 𝑤 ∈ 𝐶)) ∧ (𝜃 → ((𝑤𝑅𝑧 → ¬ 𝑤 ∈ 𝐶) → (𝑤𝑅𝑧 → ¬ 𝑤 ∈ 𝐵)))))))
23	22	exbii 1881	. . . . 5 ⊢ (∃𝑧∀𝑤((𝜑 ∧ 𝜓) → (𝑧 ∈ 𝐶 ∧ (𝜃 → (𝑤𝑅𝑧 → ¬ 𝑤 ∈ 𝐶)))) ↔ ∃𝑧∀𝑤((𝜑 ∧ 𝜓) → (𝑧 ∈ 𝐶 ∧ ((𝜃 → (𝑤𝑅𝑧 → ¬ 𝑤 ∈ 𝐶)) ∧ (𝜃 → ((𝑤𝑅𝑧 → ¬ 𝑤 ∈ 𝐶) → (𝑤𝑅𝑧 → ¬ 𝑤 ∈ 𝐵)))))))
24	1, 23	mpbi 233	. . . 4 ⊢ ∃𝑧∀𝑤((𝜑 ∧ 𝜓) → (𝑧 ∈ 𝐶 ∧ ((𝜃 → (𝑤𝑅𝑧 → ¬ 𝑤 ∈ 𝐶)) ∧ (𝜃 → ((𝑤𝑅𝑧 → ¬ 𝑤 ∈ 𝐶) → (𝑤𝑅𝑧 → ¬ 𝑤 ∈ 𝐵))))))
25		imdi 394	. . . . . . . 8 ⊢ ((𝜃 → ((𝑤𝑅𝑧 → ¬ 𝑤 ∈ 𝐶) → (𝑤𝑅𝑧 → ¬ 𝑤 ∈ 𝐵))) ↔ ((𝜃 → (𝑤𝑅𝑧 → ¬ 𝑤 ∈ 𝐶)) → (𝜃 → (𝑤𝑅𝑧 → ¬ 𝑤 ∈ 𝐵))))
26		pm3.35 815	. . . . . . . 8 ⊢ (((𝜃 → (𝑤𝑅𝑧 → ¬ 𝑤 ∈ 𝐶)) ∧ ((𝜃 → (𝑤𝑅𝑧 → ¬ 𝑤 ∈ 𝐶)) → (𝜃 → (𝑤𝑅𝑧 → ¬ 𝑤 ∈ 𝐵)))) → (𝜃 → (𝑤𝑅𝑧 → ¬ 𝑤 ∈ 𝐵)))
27	25, 26	sylan2b 606	. . . . . . 7 ⊢ (((𝜃 → (𝑤𝑅𝑧 → ¬ 𝑤 ∈ 𝐶)) ∧ (𝜃 → ((𝑤𝑅𝑧 → ¬ 𝑤 ∈ 𝐶) → (𝑤𝑅𝑧 → ¬ 𝑤 ∈ 𝐵)))) → (𝜃 → (𝑤𝑅𝑧 → ¬ 𝑤 ∈ 𝐵)))
28	27	anim2i 629	. . . . . 6 ⊢ ((𝑧 ∈ 𝐶 ∧ ((𝜃 → (𝑤𝑅𝑧 → ¬ 𝑤 ∈ 𝐶)) ∧ (𝜃 → ((𝑤𝑅𝑧 → ¬ 𝑤 ∈ 𝐶) → (𝑤𝑅𝑧 → ¬ 𝑤 ∈ 𝐵))))) → (𝑧 ∈ 𝐶 ∧ (𝜃 → (𝑤𝑅𝑧 → ¬ 𝑤 ∈ 𝐵))))
29	28	imim2i 17	. . . . 5 ⊢ (((𝜑 ∧ 𝜓) → (𝑧 ∈ 𝐶 ∧ ((𝜃 → (𝑤𝑅𝑧 → ¬ 𝑤 ∈ 𝐶)) ∧ (𝜃 → ((𝑤𝑅𝑧 → ¬ 𝑤 ∈ 𝐶) → (𝑤𝑅𝑧 → ¬ 𝑤 ∈ 𝐵)))))) → ((𝜑 ∧ 𝜓) → (𝑧 ∈ 𝐶 ∧ (𝜃 → (𝑤𝑅𝑧 → ¬ 𝑤 ∈ 𝐵)))))
30	29	alimi 1844	. . . 4 ⊢ (∀𝑤((𝜑 ∧ 𝜓) → (𝑧 ∈ 𝐶 ∧ ((𝜃 → (𝑤𝑅𝑧 → ¬ 𝑤 ∈ 𝐶)) ∧ (𝜃 → ((𝑤𝑅𝑧 → ¬ 𝑤 ∈ 𝐶) → (𝑤𝑅𝑧 → ¬ 𝑤 ∈ 𝐵)))))) → ∀𝑤((𝜑 ∧ 𝜓) → (𝑧 ∈ 𝐶 ∧ (𝜃 → (𝑤𝑅𝑧 → ¬ 𝑤 ∈ 𝐵)))))
31	24, 30	bnj101 35220	. . 3 ⊢ ∃𝑧∀𝑤((𝜑 ∧ 𝜓) → (𝑧 ∈ 𝐶 ∧ (𝜃 → (𝑤𝑅𝑧 → ¬ 𝑤 ∈ 𝐵))))
32		ancl 554	. . . . 5 ⊢ (((𝜑 ∧ 𝜓) → (𝑧 ∈ 𝐶 ∧ (𝜃 → (𝑤𝑅𝑧 → ¬ 𝑤 ∈ 𝐵)))) → ((𝜑 ∧ 𝜓) → ((𝜑 ∧ 𝜓) ∧ (𝑧 ∈ 𝐶 ∧ (𝜃 → (𝑤𝑅𝑧 → ¬ 𝑤 ∈ 𝐵))))))
33		bnj256 35203	. . . . 5 ⊢ ((𝜑 ∧ 𝜓 ∧ 𝑧 ∈ 𝐶 ∧ (𝜃 → (𝑤𝑅𝑧 → ¬ 𝑤 ∈ 𝐵))) ↔ ((𝜑 ∧ 𝜓) ∧ (𝑧 ∈ 𝐶 ∧ (𝜃 → (𝑤𝑅𝑧 → ¬ 𝑤 ∈ 𝐵)))))
34	32, 33	imbitrrdi 255	. . . 4 ⊢ (((𝜑 ∧ 𝜓) → (𝑧 ∈ 𝐶 ∧ (𝜃 → (𝑤𝑅𝑧 → ¬ 𝑤 ∈ 𝐵)))) → ((𝜑 ∧ 𝜓) → (𝜑 ∧ 𝜓 ∧ 𝑧 ∈ 𝐶 ∧ (𝜃 → (𝑤𝑅𝑧 → ¬ 𝑤 ∈ 𝐵)))))
35	34	alimi 1844	. . 3 ⊢ (∀𝑤((𝜑 ∧ 𝜓) → (𝑧 ∈ 𝐶 ∧ (𝜃 → (𝑤𝑅𝑧 → ¬ 𝑤 ∈ 𝐵)))) → ∀𝑤((𝜑 ∧ 𝜓) → (𝜑 ∧ 𝜓 ∧ 𝑧 ∈ 𝐶 ∧ (𝜃 → (𝑤𝑅𝑧 → ¬ 𝑤 ∈ 𝐵)))))
36	31, 35	bnj101 35220	. 2 ⊢ ∃𝑧∀𝑤((𝜑 ∧ 𝜓) → (𝜑 ∧ 𝜓 ∧ 𝑧 ∈ 𝐶 ∧ (𝜃 → (𝑤𝑅𝑧 → ¬ 𝑤 ∈ 𝐵))))
37		df-bnj17 35184	. . . . 5 ⊢ ((𝜑 ∧ 𝜓 ∧ 𝑧 ∈ 𝐶 ∧ (𝜃 → (𝑤𝑅𝑧 → ¬ 𝑤 ∈ 𝐵))) ↔ ((𝜑 ∧ 𝜓 ∧ 𝑧 ∈ 𝐶) ∧ (𝜃 → (𝑤𝑅𝑧 → ¬ 𝑤 ∈ 𝐵))))
38	37	imbi2i 339	. . . 4 ⊢ (((𝜑 ∧ 𝜓) → (𝜑 ∧ 𝜓 ∧ 𝑧 ∈ 𝐶 ∧ (𝜃 → (𝑤𝑅𝑧 → ¬ 𝑤 ∈ 𝐵)))) ↔ ((𝜑 ∧ 𝜓) → ((𝜑 ∧ 𝜓 ∧ 𝑧 ∈ 𝐶) ∧ (𝜃 → (𝑤𝑅𝑧 → ¬ 𝑤 ∈ 𝐵)))))
39	38	albii 1852	. . 3 ⊢ (∀𝑤((𝜑 ∧ 𝜓) → (𝜑 ∧ 𝜓 ∧ 𝑧 ∈ 𝐶 ∧ (𝜃 → (𝑤𝑅𝑧 → ¬ 𝑤 ∈ 𝐵)))) ↔ ∀𝑤((𝜑 ∧ 𝜓) → ((𝜑 ∧ 𝜓 ∧ 𝑧 ∈ 𝐶) ∧ (𝜃 → (𝑤𝑅𝑧 → ¬ 𝑤 ∈ 𝐵)))))
40	39	exbii 1881	. 2 ⊢ (∃𝑧∀𝑤((𝜑 ∧ 𝜓) → (𝜑 ∧ 𝜓 ∧ 𝑧 ∈ 𝐶 ∧ (𝜃 → (𝑤𝑅𝑧 → ¬ 𝑤 ∈ 𝐵)))) ↔ ∃𝑧∀𝑤((𝜑 ∧ 𝜓) → ((𝜑 ∧ 𝜓 ∧ 𝑧 ∈ 𝐶) ∧ (𝜃 → (𝑤𝑅𝑧 → ¬ 𝑤 ∈ 𝐵)))))
41	36, 40	mpbi 233	1 ⊢ ∃𝑧∀𝑤((𝜑 ∧ 𝜓) → ((𝜑 ∧ 𝜓 ∧ 𝑧 ∈ 𝐶) ∧ (𝜃 → (𝑤𝑅𝑧 → ¬ 𝑤 ∈ 𝐵))))

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ∧ wa 401   ∨ wo 861   ∧ w3a 1103  ∀wal 1568   = wceq 1570  ∃wex 1812   ∈ wcel 2145   ∩ cin 3901   class class class wbr 5107   ∧ w-bnj17 35183   trClc-bnj18 35191

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-ext 2734

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-ex 1813  df-sb 2100  df-clab 2741  df-cleq 2754  df-clel 2837  df-v 3455  df-in 3909  df-bnj17 35184

This theorem is used by:  bnj1190  35504