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Theorem bnj1172 31401
Description: Technical lemma for bnj69 31410. 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 556 . . . . . 6 (((𝜑𝜓𝑧𝐶) ∧ (𝜃 → (𝑤𝑅𝑧 → ¬ 𝑤𝐵))) ↔ ((𝜑𝜓𝑧𝐶) ∧ (𝑤𝐴 → (𝑤𝑅𝑧 → ¬ 𝑤𝐵))))
54imbi2i 325 . . . . 5 (((𝜑𝜓) → ((𝜑𝜓𝑧𝐶) ∧ (𝜃 → (𝑤𝑅𝑧 → ¬ 𝑤𝐵)))) ↔ ((𝜑𝜓) → ((𝜑𝜓𝑧𝐶) ∧ (𝑤𝐴 → (𝑤𝑅𝑧 → ¬ 𝑤𝐵)))))
65albii 1894 . . . 4 (∀𝑤((𝜑𝜓) → ((𝜑𝜓𝑧𝐶) ∧ (𝜃 → (𝑤𝑅𝑧 → ¬ 𝑤𝐵)))) ↔ ∀𝑤((𝜑𝜓) → ((𝜑𝜓𝑧𝐶) ∧ (𝑤𝐴 → (𝑤𝑅𝑧 → ¬ 𝑤𝐵)))))
76exbii 1923 . . 3 (∃𝑧𝑤((𝜑𝜓) → ((𝜑𝜓𝑧𝐶) ∧ (𝜃 → (𝑤𝑅𝑧 → ¬ 𝑤𝐵)))) ↔ ∃𝑧𝑤((𝜑𝜓) → ((𝜑𝜓𝑧𝐶) ∧ (𝑤𝐴 → (𝑤𝑅𝑧 → ¬ 𝑤𝐵)))))
81, 7mpbi 220 . 2 𝑧𝑤((𝜑𝜓) → ((𝜑𝜓𝑧𝐶) ∧ (𝑤𝐴 → (𝑤𝑅𝑧 → ¬ 𝑤𝐵))))
9 simp3 1131 . . . . . . 7 ((𝜑𝜓𝑧𝐶) → 𝑧𝐶)
10 bnj1172.3 . . . . . . 7 𝐶 = ( trCl(𝑋, 𝐴, 𝑅) ∩ 𝐵)
119, 10syl6eleq 2859 . . . . . 6 ((𝜑𝜓𝑧𝐶) → 𝑧 ∈ ( trCl(𝑋, 𝐴, 𝑅) ∩ 𝐵))
1211elin2d 3952 . . . . 5 ((𝜑𝜓𝑧𝐶) → 𝑧𝐵)
1312anim1i 594 . . . 4 (((𝜑𝜓𝑧𝐶) ∧ (𝑤𝐴 → (𝑤𝑅𝑧 → ¬ 𝑤𝐵))) → (𝑧𝐵 ∧ (𝑤𝐴 → (𝑤𝑅𝑧 → ¬ 𝑤𝐵))))
1413imim2i 16 . . 3 (((𝜑𝜓) → ((𝜑𝜓𝑧𝐶) ∧ (𝑤𝐴 → (𝑤𝑅𝑧 → ¬ 𝑤𝐵)))) → ((𝜑𝜓) → (𝑧𝐵 ∧ (𝑤𝐴 → (𝑤𝑅𝑧 → ¬ 𝑤𝐵)))))
1514alimi 1886 . 2 (∀𝑤((𝜑𝜓) → ((𝜑𝜓𝑧𝐶) ∧ (𝑤𝐴 → (𝑤𝑅𝑧 → ¬ 𝑤𝐵)))) → ∀𝑤((𝜑𝜓) → (𝑧𝐵 ∧ (𝑤𝐴 → (𝑤𝑅𝑧 → ¬ 𝑤𝐵)))))
168, 15bnj101 31123 1 𝑧𝑤((𝜑𝜓) → (𝑧𝐵 ∧ (𝑤𝐴 → (𝑤𝑅𝑧 → ¬ 𝑤𝐵))))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 196  wa 382  w3a 1070  wal 1628   = wceq 1630  wex 1851  wcel 2144  cin 3720   class class class wbr 4784   trClc-bnj18 31094
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1869  ax-4 1884  ax-5 1990  ax-6 2056  ax-7 2092  ax-9 2153  ax-10 2173  ax-11 2189  ax-12 2202  ax-13 2407  ax-ext 2750
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 827  df-3an 1072  df-tru 1633  df-ex 1852  df-nf 1857  df-sb 2049  df-clab 2757  df-cleq 2763  df-clel 2766  df-nfc 2901  df-v 3351  df-in 3728
This theorem is referenced by:  bnj1190  31408
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