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Theorem bnj1172 30323
 Description: Technical lemma for bnj69 30332. 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 𝑧𝑤((𝜑𝜓) → ((𝜑𝜓𝑧𝐶) ∧ (𝜃 → (𝑤𝑅𝑧 → ¬ 𝑤𝐵))))
bnj1172.113 ((𝜑𝜓𝑧𝐶) → (𝜃𝑤𝐴))
Assertion
Ref Expression
bnj1172 𝑧𝑤((𝜑𝜓) → (𝑧𝐵 ∧ (𝑤𝐴 → (𝑤𝑅𝑧 → ¬ 𝑤𝐵))))

Proof of Theorem bnj1172
StepHypRef Expression
1 bnj1172.96 . . 3 𝑧𝑤((𝜑𝜓) → ((𝜑𝜓𝑧𝐶) ∧ (𝜃 → (𝑤𝑅𝑧 → ¬ 𝑤𝐵))))
2 bnj1172.113 . . . . . . . 8 ((𝜑𝜓𝑧𝐶) → (𝜃𝑤𝐴))
32imbi1d 330 . . . . . . 7 ((𝜑𝜓𝑧𝐶) → ((𝜃 → (𝑤𝑅𝑧 → ¬ 𝑤𝐵)) ↔ (𝑤𝐴 → (𝑤𝑅𝑧 → ¬ 𝑤𝐵))))
43pm5.32i 667 . . . . . 6 (((𝜑𝜓𝑧𝐶) ∧ (𝜃 → (𝑤𝑅𝑧 → ¬ 𝑤𝐵))) ↔ ((𝜑𝜓𝑧𝐶) ∧ (𝑤𝐴 → (𝑤𝑅𝑧 → ¬ 𝑤𝐵))))
54imbi2i 325 . . . . 5 (((𝜑𝜓) → ((𝜑𝜓𝑧𝐶) ∧ (𝜃 → (𝑤𝑅𝑧 → ¬ 𝑤𝐵)))) ↔ ((𝜑𝜓) → ((𝜑𝜓𝑧𝐶) ∧ (𝑤𝐴 → (𝑤𝑅𝑧 → ¬ 𝑤𝐵)))))
65albii 1737 . . . 4 (∀𝑤((𝜑𝜓) → ((𝜑𝜓𝑧𝐶) ∧ (𝜃 → (𝑤𝑅𝑧 → ¬ 𝑤𝐵)))) ↔ ∀𝑤((𝜑𝜓) → ((𝜑𝜓𝑧𝐶) ∧ (𝑤𝐴 → (𝑤𝑅𝑧 → ¬ 𝑤𝐵)))))
76exbii 1764 . . 3 (∃𝑧𝑤((𝜑𝜓) → ((𝜑𝜓𝑧𝐶) ∧ (𝜃 → (𝑤𝑅𝑧 → ¬ 𝑤𝐵)))) ↔ ∃𝑧𝑤((𝜑𝜓) → ((𝜑𝜓𝑧𝐶) ∧ (𝑤𝐴 → (𝑤𝑅𝑧 → ¬ 𝑤𝐵)))))
81, 7mpbi 219 . 2 𝑧𝑤((𝜑𝜓) → ((𝜑𝜓𝑧𝐶) ∧ (𝑤𝐴 → (𝑤𝑅𝑧 → ¬ 𝑤𝐵))))
9 simp3 1056 . . . . . . 7 ((𝜑𝜓𝑧𝐶) → 𝑧𝐶)
10 bnj1172.3 . . . . . . 7 𝐶 = ( trCl(𝑋, 𝐴, 𝑅) ∩ 𝐵)
119, 10syl6eleq 2698 . . . . . 6 ((𝜑𝜓𝑧𝐶) → 𝑧 ∈ ( trCl(𝑋, 𝐴, 𝑅) ∩ 𝐵))
12 elin 3758 . . . . . . 7 (𝑧 ∈ ( trCl(𝑋, 𝐴, 𝑅) ∩ 𝐵) ↔ (𝑧 ∈ trCl(𝑋, 𝐴, 𝑅) ∧ 𝑧𝐵))
1312simprbi 479 . . . . . 6 (𝑧 ∈ ( trCl(𝑋, 𝐴, 𝑅) ∩ 𝐵) → 𝑧𝐵)
1411, 13syl 17 . . . . 5 ((𝜑𝜓𝑧𝐶) → 𝑧𝐵)
1514anim1i 590 . . . 4 (((𝜑𝜓𝑧𝐶) ∧ (𝑤𝐴 → (𝑤𝑅𝑧 → ¬ 𝑤𝐵))) → (𝑧𝐵 ∧ (𝑤𝐴 → (𝑤𝑅𝑧 → ¬ 𝑤𝐵))))
1615imim2i 16 . . 3 (((𝜑𝜓) → ((𝜑𝜓𝑧𝐶) ∧ (𝑤𝐴 → (𝑤𝑅𝑧 → ¬ 𝑤𝐵)))) → ((𝜑𝜓) → (𝑧𝐵 ∧ (𝑤𝐴 → (𝑤𝑅𝑧 → ¬ 𝑤𝐵)))))
1716alimi 1730 . 2 (∀𝑤((𝜑𝜓) → ((𝜑𝜓𝑧𝐶) ∧ (𝑤𝐴 → (𝑤𝑅𝑧 → ¬ 𝑤𝐵)))) → ∀𝑤((𝜑𝜓) → (𝑧𝐵 ∧ (𝑤𝐴 → (𝑤𝑅𝑧 → ¬ 𝑤𝐵)))))
188, 17bnj101 30043 1 𝑧𝑤((𝜑𝜓) → (𝑧𝐵 ∧ (𝑤𝐴 → (𝑤𝑅𝑧 → ¬ 𝑤𝐵))))
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 195   ∧ wa 383   ∧ w3a 1031  ∀wal 1473   = wceq 1475  ∃wex 1695   ∈ wcel 1977   ∩ cin 3539   class class class wbr 4583   trClc-bnj18 30013 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-v 3175  df-in 3547 This theorem is referenced by:  bnj1190  30330
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