Theorem bnj1172 35015

Description: Technical lemma for bnj69 35024. 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
bnj1172.3	⊢ 𝐶 = ( trCl(𝑋, 𝐴, 𝑅) ∩ 𝐵)
bnj1172.96	⊢ ∃𝑧∀𝑤((𝜑 ∧ 𝜓) → ((𝜑 ∧ 𝜓 ∧ 𝑧 ∈ 𝐶) ∧ (𝜃 → (𝑤𝑅𝑧 → ¬ 𝑤 ∈ 𝐵))))
bnj1172.113	⊢ ((𝜑 ∧ 𝜓 ∧ 𝑧 ∈ 𝐶) → (𝜃 ↔ 𝑤 ∈ 𝐴))

Assertion

Ref	Expression
bnj1172	⊢ ∃𝑧∀𝑤((𝜑 ∧ 𝜓) → (𝑧 ∈ 𝐵 ∧ (𝑤 ∈ 𝐴 → (𝑤𝑅𝑧 → ¬ 𝑤 ∈ 𝐵))))

Proof of Theorem bnj1172

Step	Hyp	Ref	Expression
1		bnj1172.96	. . 3 ⊢ ∃𝑧∀𝑤((𝜑 ∧ 𝜓) → ((𝜑 ∧ 𝜓 ∧ 𝑧 ∈ 𝐶) ∧ (𝜃 → (𝑤𝑅𝑧 → ¬ 𝑤 ∈ 𝐵))))
2		bnj1172.113	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝜓 ∧ 𝑧 ∈ 𝐶) → (𝜃 ↔ 𝑤 ∈ 𝐴))
3	2	imbi1d 341	. . . . . . 7 ⊢ ((𝜑 ∧ 𝜓 ∧ 𝑧 ∈ 𝐶) → ((𝜃 → (𝑤𝑅𝑧 → ¬ 𝑤 ∈ 𝐵)) ↔ (𝑤 ∈ 𝐴 → (𝑤𝑅𝑧 → ¬ 𝑤 ∈ 𝐵))))
4	3	pm5.32i 574	. . . . . 6 ⊢ (((𝜑 ∧ 𝜓 ∧ 𝑧 ∈ 𝐶) ∧ (𝜃 → (𝑤𝑅𝑧 → ¬ 𝑤 ∈ 𝐵))) ↔ ((𝜑 ∧ 𝜓 ∧ 𝑧 ∈ 𝐶) ∧ (𝑤 ∈ 𝐴 → (𝑤𝑅𝑧 → ¬ 𝑤 ∈ 𝐵))))
5	4	imbi2i 336	. . . . 5 ⊢ (((𝜑 ∧ 𝜓) → ((𝜑 ∧ 𝜓 ∧ 𝑧 ∈ 𝐶) ∧ (𝜃 → (𝑤𝑅𝑧 → ¬ 𝑤 ∈ 𝐵)))) ↔ ((𝜑 ∧ 𝜓) → ((𝜑 ∧ 𝜓 ∧ 𝑧 ∈ 𝐶) ∧ (𝑤 ∈ 𝐴 → (𝑤𝑅𝑧 → ¬ 𝑤 ∈ 𝐵)))))
6	5	albii 1819	. . . 4 ⊢ (∀𝑤((𝜑 ∧ 𝜓) → ((𝜑 ∧ 𝜓 ∧ 𝑧 ∈ 𝐶) ∧ (𝜃 → (𝑤𝑅𝑧 → ¬ 𝑤 ∈ 𝐵)))) ↔ ∀𝑤((𝜑 ∧ 𝜓) → ((𝜑 ∧ 𝜓 ∧ 𝑧 ∈ 𝐶) ∧ (𝑤 ∈ 𝐴 → (𝑤𝑅𝑧 → ¬ 𝑤 ∈ 𝐵)))))
7	6	exbii 1848	. . 3 ⊢ (∃𝑧∀𝑤((𝜑 ∧ 𝜓) → ((𝜑 ∧ 𝜓 ∧ 𝑧 ∈ 𝐶) ∧ (𝜃 → (𝑤𝑅𝑧 → ¬ 𝑤 ∈ 𝐵)))) ↔ ∃𝑧∀𝑤((𝜑 ∧ 𝜓) → ((𝜑 ∧ 𝜓 ∧ 𝑧 ∈ 𝐶) ∧ (𝑤 ∈ 𝐴 → (𝑤𝑅𝑧 → ¬ 𝑤 ∈ 𝐵)))))
8	1, 7	mpbi 230	. 2 ⊢ ∃𝑧∀𝑤((𝜑 ∧ 𝜓) → ((𝜑 ∧ 𝜓 ∧ 𝑧 ∈ 𝐶) ∧ (𝑤 ∈ 𝐴 → (𝑤𝑅𝑧 → ¬ 𝑤 ∈ 𝐵))))
9		simp3 1139	. . . . . . 7 ⊢ ((𝜑 ∧ 𝜓 ∧ 𝑧 ∈ 𝐶) → 𝑧 ∈ 𝐶)
10		bnj1172.3	. . . . . . 7 ⊢ 𝐶 = ( trCl(𝑋, 𝐴, 𝑅) ∩ 𝐵)
11	9, 10	eleqtrdi 2851	. . . . . 6 ⊢ ((𝜑 ∧ 𝜓 ∧ 𝑧 ∈ 𝐶) → 𝑧 ∈ ( trCl(𝑋, 𝐴, 𝑅) ∩ 𝐵))
12	11	elin2d 4205	. . . . 5 ⊢ ((𝜑 ∧ 𝜓 ∧ 𝑧 ∈ 𝐶) → 𝑧 ∈ 𝐵)
13	12	anim1i 615	. . . 4 ⊢ (((𝜑 ∧ 𝜓 ∧ 𝑧 ∈ 𝐶) ∧ (𝑤 ∈ 𝐴 → (𝑤𝑅𝑧 → ¬ 𝑤 ∈ 𝐵))) → (𝑧 ∈ 𝐵 ∧ (𝑤 ∈ 𝐴 → (𝑤𝑅𝑧 → ¬ 𝑤 ∈ 𝐵))))
14	13	imim2i 16	. . 3 ⊢ (((𝜑 ∧ 𝜓) → ((𝜑 ∧ 𝜓 ∧ 𝑧 ∈ 𝐶) ∧ (𝑤 ∈ 𝐴 → (𝑤𝑅𝑧 → ¬ 𝑤 ∈ 𝐵)))) → ((𝜑 ∧ 𝜓) → (𝑧 ∈ 𝐵 ∧ (𝑤 ∈ 𝐴 → (𝑤𝑅𝑧 → ¬ 𝑤 ∈ 𝐵)))))
15	14	alimi 1811	. 2 ⊢ (∀𝑤((𝜑 ∧ 𝜓) → ((𝜑 ∧ 𝜓 ∧ 𝑧 ∈ 𝐶) ∧ (𝑤 ∈ 𝐴 → (𝑤𝑅𝑧 → ¬ 𝑤 ∈ 𝐵)))) → ∀𝑤((𝜑 ∧ 𝜓) → (𝑧 ∈ 𝐵 ∧ (𝑤 ∈ 𝐴 → (𝑤𝑅𝑧 → ¬ 𝑤 ∈ 𝐵)))))
16	8, 15	bnj101 34737	1 ⊢ ∃𝑧∀𝑤((𝜑 ∧ 𝜓) → (𝑧 ∈ 𝐵 ∧ (𝑤 ∈ 𝐴 → (𝑤𝑅𝑧 → ¬ 𝑤 ∈ 𝐵))))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 206   ∧ wa 395   ∧ w3a 1087  ∀wal 1538   = wceq 1540  ∃wex 1779   ∈ wcel 2108   ∩ cin 3950   class class class wbr 5143   trClc-bnj18 34708

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2708

This theorem depends on definitions:  df-bi 207  df-an 396  df-3an 1089  df-tru 1543  df-ex 1780  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-v 3482  df-in 3958

This theorem is referenced by:  bnj1190  35022