Theorem bnj1171 32647

Description: Technical lemma for bnj69 32657. 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
bnj1171.13	⊢ ((𝜑 ∧ 𝜓) → 𝐵 ⊆ 𝐴)
bnj1171.129	⊢ ∃𝑧∀𝑤((𝜑 ∧ 𝜓) → (𝑧 ∈ 𝐵 ∧ (𝑤 ∈ 𝐴 → (𝑤𝑅𝑧 → ¬ 𝑤 ∈ 𝐵))))

Assertion

Ref	Expression
bnj1171	⊢ ∃𝑧∀𝑤((𝜑 ∧ 𝜓) → (𝑧 ∈ 𝐵 ∧ (𝑤 ∈ 𝐵 → ¬ 𝑤𝑅𝑧)))

Proof of Theorem bnj1171

Step	Hyp	Ref	Expression
1		bnj1171.129	. 2 ⊢ ∃𝑧∀𝑤((𝜑 ∧ 𝜓) → (𝑧 ∈ 𝐵 ∧ (𝑤 ∈ 𝐴 → (𝑤𝑅𝑧 → ¬ 𝑤 ∈ 𝐵))))
2		bnj1171.13	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝜓) → 𝐵 ⊆ 𝐴)
3	2	sseld 3886	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝜓) → (𝑤 ∈ 𝐵 → 𝑤 ∈ 𝐴))
4	3	pm4.71rd 566	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝜓) → (𝑤 ∈ 𝐵 ↔ (𝑤 ∈ 𝐴 ∧ 𝑤 ∈ 𝐵)))
5	4	imbi1d 345	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝜓) → ((𝑤 ∈ 𝐵 → ¬ 𝑤𝑅𝑧) ↔ ((𝑤 ∈ 𝐴 ∧ 𝑤 ∈ 𝐵) → ¬ 𝑤𝑅𝑧)))
6		impexp 454	. . . . . . . 8 ⊢ (((𝑤 ∈ 𝐴 ∧ 𝑤 ∈ 𝐵) → ¬ 𝑤𝑅𝑧) ↔ (𝑤 ∈ 𝐴 → (𝑤 ∈ 𝐵 → ¬ 𝑤𝑅𝑧)))
7	5, 6	bitrdi 290	. . . . . . 7 ⊢ ((𝜑 ∧ 𝜓) → ((𝑤 ∈ 𝐵 → ¬ 𝑤𝑅𝑧) ↔ (𝑤 ∈ 𝐴 → (𝑤 ∈ 𝐵 → ¬ 𝑤𝑅𝑧))))
8		con2b 363	. . . . . . . 8 ⊢ ((𝑤𝑅𝑧 → ¬ 𝑤 ∈ 𝐵) ↔ (𝑤 ∈ 𝐵 → ¬ 𝑤𝑅𝑧))
9	8	imbi2i 339	. . . . . . 7 ⊢ ((𝑤 ∈ 𝐴 → (𝑤𝑅𝑧 → ¬ 𝑤 ∈ 𝐵)) ↔ (𝑤 ∈ 𝐴 → (𝑤 ∈ 𝐵 → ¬ 𝑤𝑅𝑧)))
10	7, 9	bitr4di 292	. . . . . 6 ⊢ ((𝜑 ∧ 𝜓) → ((𝑤 ∈ 𝐵 → ¬ 𝑤𝑅𝑧) ↔ (𝑤 ∈ 𝐴 → (𝑤𝑅𝑧 → ¬ 𝑤 ∈ 𝐵))))
11	10	anbi2d 632	. . . . 5 ⊢ ((𝜑 ∧ 𝜓) → ((𝑧 ∈ 𝐵 ∧ (𝑤 ∈ 𝐵 → ¬ 𝑤𝑅𝑧)) ↔ (𝑧 ∈ 𝐵 ∧ (𝑤 ∈ 𝐴 → (𝑤𝑅𝑧 → ¬ 𝑤 ∈ 𝐵)))))
12	11	pm5.74i 274	. . . 4 ⊢ (((𝜑 ∧ 𝜓) → (𝑧 ∈ 𝐵 ∧ (𝑤 ∈ 𝐵 → ¬ 𝑤𝑅𝑧))) ↔ ((𝜑 ∧ 𝜓) → (𝑧 ∈ 𝐵 ∧ (𝑤 ∈ 𝐴 → (𝑤𝑅𝑧 → ¬ 𝑤 ∈ 𝐵)))))
13	12	albii 1827	. . 3 ⊢ (∀𝑤((𝜑 ∧ 𝜓) → (𝑧 ∈ 𝐵 ∧ (𝑤 ∈ 𝐵 → ¬ 𝑤𝑅𝑧))) ↔ ∀𝑤((𝜑 ∧ 𝜓) → (𝑧 ∈ 𝐵 ∧ (𝑤 ∈ 𝐴 → (𝑤𝑅𝑧 → ¬ 𝑤 ∈ 𝐵)))))
14	13	exbii 1855	. 2 ⊢ (∃𝑧∀𝑤((𝜑 ∧ 𝜓) → (𝑧 ∈ 𝐵 ∧ (𝑤 ∈ 𝐵 → ¬ 𝑤𝑅𝑧))) ↔ ∃𝑧∀𝑤((𝜑 ∧ 𝜓) → (𝑧 ∈ 𝐵 ∧ (𝑤 ∈ 𝐴 → (𝑤𝑅𝑧 → ¬ 𝑤 ∈ 𝐵)))))
15	1, 14	mpbir 234	1 ⊢ ∃𝑧∀𝑤((𝜑 ∧ 𝜓) → (𝑧 ∈ 𝐵 ∧ (𝑤 ∈ 𝐵 → ¬ 𝑤𝑅𝑧)))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 399  ∀wal 1541  ∃wex 1787   ∈ wcel 2112   ⊆ wss 3853   class class class wbr 5039

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2018  ax-8 2114  ax-9 2122  ax-ext 2708

This theorem depends on definitions:  df-bi 210  df-an 400  df-tru 1546  df-ex 1788  df-sb 2073  df-clab 2715  df-cleq 2728  df-clel 2809  df-v 3400  df-in 3860  df-ss 3870

This theorem is referenced by:  bnj1190  32655