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Theorem bnj1171 32386
 Description: Technical lemma for bnj69 32396. 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 ((𝜑𝜓) → 𝐵𝐴)
bnj1171.129 𝑧𝑤((𝜑𝜓) → (𝑧𝐵 ∧ (𝑤𝐴 → (𝑤𝑅𝑧 → ¬ 𝑤𝐵))))
Assertion
Ref Expression
bnj1171 𝑧𝑤((𝜑𝜓) → (𝑧𝐵 ∧ (𝑤𝐵 → ¬ 𝑤𝑅𝑧)))

Proof of Theorem bnj1171
StepHypRef Expression
1 bnj1171.129 . 2 𝑧𝑤((𝜑𝜓) → (𝑧𝐵 ∧ (𝑤𝐴 → (𝑤𝑅𝑧 → ¬ 𝑤𝐵))))
2 bnj1171.13 . . . . . . . . . . 11 ((𝜑𝜓) → 𝐵𝐴)
32sseld 3917 . . . . . . . . . 10 ((𝜑𝜓) → (𝑤𝐵𝑤𝐴))
43pm4.71rd 566 . . . . . . . . 9 ((𝜑𝜓) → (𝑤𝐵 ↔ (𝑤𝐴𝑤𝐵)))
54imbi1d 345 . . . . . . . 8 ((𝜑𝜓) → ((𝑤𝐵 → ¬ 𝑤𝑅𝑧) ↔ ((𝑤𝐴𝑤𝐵) → ¬ 𝑤𝑅𝑧)))
6 impexp 454 . . . . . . . 8 (((𝑤𝐴𝑤𝐵) → ¬ 𝑤𝑅𝑧) ↔ (𝑤𝐴 → (𝑤𝐵 → ¬ 𝑤𝑅𝑧)))
75, 6syl6bb 290 . . . . . . 7 ((𝜑𝜓) → ((𝑤𝐵 → ¬ 𝑤𝑅𝑧) ↔ (𝑤𝐴 → (𝑤𝐵 → ¬ 𝑤𝑅𝑧))))
8 con2b 363 . . . . . . . 8 ((𝑤𝑅𝑧 → ¬ 𝑤𝐵) ↔ (𝑤𝐵 → ¬ 𝑤𝑅𝑧))
98imbi2i 339 . . . . . . 7 ((𝑤𝐴 → (𝑤𝑅𝑧 → ¬ 𝑤𝐵)) ↔ (𝑤𝐴 → (𝑤𝐵 → ¬ 𝑤𝑅𝑧)))
107, 9syl6bbr 292 . . . . . 6 ((𝜑𝜓) → ((𝑤𝐵 → ¬ 𝑤𝑅𝑧) ↔ (𝑤𝐴 → (𝑤𝑅𝑧 → ¬ 𝑤𝐵))))
1110anbi2d 631 . . . . 5 ((𝜑𝜓) → ((𝑧𝐵 ∧ (𝑤𝐵 → ¬ 𝑤𝑅𝑧)) ↔ (𝑧𝐵 ∧ (𝑤𝐴 → (𝑤𝑅𝑧 → ¬ 𝑤𝐵)))))
1211pm5.74i 274 . . . 4 (((𝜑𝜓) → (𝑧𝐵 ∧ (𝑤𝐵 → ¬ 𝑤𝑅𝑧))) ↔ ((𝜑𝜓) → (𝑧𝐵 ∧ (𝑤𝐴 → (𝑤𝑅𝑧 → ¬ 𝑤𝐵)))))
1312albii 1821 . . 3 (∀𝑤((𝜑𝜓) → (𝑧𝐵 ∧ (𝑤𝐵 → ¬ 𝑤𝑅𝑧))) ↔ ∀𝑤((𝜑𝜓) → (𝑧𝐵 ∧ (𝑤𝐴 → (𝑤𝑅𝑧 → ¬ 𝑤𝐵)))))
1413exbii 1849 . 2 (∃𝑧𝑤((𝜑𝜓) → (𝑧𝐵 ∧ (𝑤𝐵 → ¬ 𝑤𝑅𝑧))) ↔ ∃𝑧𝑤((𝜑𝜓) → (𝑧𝐵 ∧ (𝑤𝐴 → (𝑤𝑅𝑧 → ¬ 𝑤𝐵)))))
151, 14mpbir 234 1 𝑧𝑤((𝜑𝜓) → (𝑧𝐵 ∧ (𝑤𝐵 → ¬ 𝑤𝑅𝑧)))
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 399  ∀wal 1536  ∃wex 1781   ∈ wcel 2112   ⊆ wss 3884   class class class wbr 5033 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2114  ax-9 2122  ax-ext 2773 This theorem depends on definitions:  df-bi 210  df-an 400  df-ex 1782  df-sb 2070  df-clab 2780  df-cleq 2794  df-clel 2873  df-v 3446  df-in 3891  df-ss 3901 This theorem is referenced by:  bnj1190  32394
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