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Theorem bnj1174 32275
Description: Technical lemma for bnj69 32282. 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
bnj1174.3 𝐶 = ( trCl(𝑋, 𝐴, 𝑅) ∩ 𝐵)
bnj1174.59 𝑧𝑤((𝜑𝜓) → (𝑧𝐶 ∧ (𝜃 → (𝑤𝑅𝑧 → ¬ 𝑤𝐶))))
bnj1174.74 (𝜃 → (𝑤𝑅𝑧𝑤 ∈ trCl(𝑋, 𝐴, 𝑅)))
Assertion
Ref Expression
bnj1174 𝑧𝑤((𝜑𝜓) → ((𝜑𝜓𝑧𝐶) ∧ (𝜃 → (𝑤𝑅𝑧 → ¬ 𝑤𝐵))))

Proof of Theorem bnj1174
StepHypRef Expression
1 bnj1174.59 . . . . 5 𝑧𝑤((𝜑𝜓) → (𝑧𝐶 ∧ (𝜃 → (𝑤𝑅𝑧 → ¬ 𝑤𝐶))))
2 bnj1174.74 . . . . . . . . . . . 12 (𝜃 → (𝑤𝑅𝑧𝑤 ∈ trCl(𝑋, 𝐴, 𝑅)))
3 bnj1174.3 . . . . . . . . . . . . . . . 16 𝐶 = ( trCl(𝑋, 𝐴, 𝑅) ∩ 𝐵)
43eleq2i 2904 . . . . . . . . . . . . . . 15 (𝑤𝐶𝑤 ∈ ( trCl(𝑋, 𝐴, 𝑅) ∩ 𝐵))
54notbii 322 . . . . . . . . . . . . . 14 𝑤𝐶 ↔ ¬ 𝑤 ∈ ( trCl(𝑋, 𝐴, 𝑅) ∩ 𝐵))
6 ianor 978 . . . . . . . . . . . . . . . . 17 (¬ (𝑤 ∈ trCl(𝑋, 𝐴, 𝑅) ∧ 𝑤𝐵) ↔ (¬ 𝑤 ∈ trCl(𝑋, 𝐴, 𝑅) ∨ ¬ 𝑤𝐵))
7 elin 4168 . . . . . . . . . . . . . . . . . 18 (𝑤 ∈ ( trCl(𝑋, 𝐴, 𝑅) ∩ 𝐵) ↔ (𝑤 ∈ trCl(𝑋, 𝐴, 𝑅) ∧ 𝑤𝐵))
87notbii 322 . . . . . . . . . . . . . . . . 17 𝑤 ∈ ( trCl(𝑋, 𝐴, 𝑅) ∩ 𝐵) ↔ ¬ (𝑤 ∈ trCl(𝑋, 𝐴, 𝑅) ∧ 𝑤𝐵))
9 pm4.62 852 . . . . . . . . . . . . . . . . 17 ((𝑤 ∈ trCl(𝑋, 𝐴, 𝑅) → ¬ 𝑤𝐵) ↔ (¬ 𝑤 ∈ trCl(𝑋, 𝐴, 𝑅) ∨ ¬ 𝑤𝐵))
106, 8, 93bitr4i 305 . . . . . . . . . . . . . . . 16 𝑤 ∈ ( trCl(𝑋, 𝐴, 𝑅) ∩ 𝐵) ↔ (𝑤 ∈ trCl(𝑋, 𝐴, 𝑅) → ¬ 𝑤𝐵))
1110biimpi 218 . . . . . . . . . . . . . . 15 𝑤 ∈ ( trCl(𝑋, 𝐴, 𝑅) ∩ 𝐵) → (𝑤 ∈ trCl(𝑋, 𝐴, 𝑅) → ¬ 𝑤𝐵))
1211impcom 410 . . . . . . . . . . . . . 14 ((𝑤 ∈ trCl(𝑋, 𝐴, 𝑅) ∧ ¬ 𝑤 ∈ ( trCl(𝑋, 𝐴, 𝑅) ∩ 𝐵)) → ¬ 𝑤𝐵)
135, 12sylan2b 595 . . . . . . . . . . . . 13 ((𝑤 ∈ trCl(𝑋, 𝐴, 𝑅) ∧ ¬ 𝑤𝐶) → ¬ 𝑤𝐵)
1413ex 415 . . . . . . . . . . . 12 (𝑤 ∈ trCl(𝑋, 𝐴, 𝑅) → (¬ 𝑤𝐶 → ¬ 𝑤𝐵))
152, 14syl6 35 . . . . . . . . . . 11 (𝜃 → (𝑤𝑅𝑧 → (¬ 𝑤𝐶 → ¬ 𝑤𝐵)))
1615a2d 29 . . . . . . . . . 10 (𝜃 → ((𝑤𝑅𝑧 → ¬ 𝑤𝐶) → (𝑤𝑅𝑧 → ¬ 𝑤𝐵)))
1716biantru 532 . . . . . . . . 9 ((𝑧𝐶 ∧ (𝜃 → (𝑤𝑅𝑧 → ¬ 𝑤𝐶))) ↔ ((𝑧𝐶 ∧ (𝜃 → (𝑤𝑅𝑧 → ¬ 𝑤𝐶))) ∧ (𝜃 → ((𝑤𝑅𝑧 → ¬ 𝑤𝐶) → (𝑤𝑅𝑧 → ¬ 𝑤𝐵)))))
18 df-3an 1085 . . . . . . . . 9 ((𝑧𝐶 ∧ (𝜃 → (𝑤𝑅𝑧 → ¬ 𝑤𝐶)) ∧ (𝜃 → ((𝑤𝑅𝑧 → ¬ 𝑤𝐶) → (𝑤𝑅𝑧 → ¬ 𝑤𝐵)))) ↔ ((𝑧𝐶 ∧ (𝜃 → (𝑤𝑅𝑧 → ¬ 𝑤𝐶))) ∧ (𝜃 → ((𝑤𝑅𝑧 → ¬ 𝑤𝐶) → (𝑤𝑅𝑧 → ¬ 𝑤𝐵)))))
19 3anass 1091 . . . . . . . . 9 ((𝑧𝐶 ∧ (𝜃 → (𝑤𝑅𝑧 → ¬ 𝑤𝐶)) ∧ (𝜃 → ((𝑤𝑅𝑧 → ¬ 𝑤𝐶) → (𝑤𝑅𝑧 → ¬ 𝑤𝐵)))) ↔ (𝑧𝐶 ∧ ((𝜃 → (𝑤𝑅𝑧 → ¬ 𝑤𝐶)) ∧ (𝜃 → ((𝑤𝑅𝑧 → ¬ 𝑤𝐶) → (𝑤𝑅𝑧 → ¬ 𝑤𝐵))))))
2017, 18, 193bitr2i 301 . . . . . . . 8 ((𝑧𝐶 ∧ (𝜃 → (𝑤𝑅𝑧 → ¬ 𝑤𝐶))) ↔ (𝑧𝐶 ∧ ((𝜃 → (𝑤𝑅𝑧 → ¬ 𝑤𝐶)) ∧ (𝜃 → ((𝑤𝑅𝑧 → ¬ 𝑤𝐶) → (𝑤𝑅𝑧 → ¬ 𝑤𝐵))))))
2120imbi2i 338 . . . . . . 7 (((𝜑𝜓) → (𝑧𝐶 ∧ (𝜃 → (𝑤𝑅𝑧 → ¬ 𝑤𝐶)))) ↔ ((𝜑𝜓) → (𝑧𝐶 ∧ ((𝜃 → (𝑤𝑅𝑧 → ¬ 𝑤𝐶)) ∧ (𝜃 → ((𝑤𝑅𝑧 → ¬ 𝑤𝐶) → (𝑤𝑅𝑧 → ¬ 𝑤𝐵)))))))
2221albii 1816 . . . . . 6 (∀𝑤((𝜑𝜓) → (𝑧𝐶 ∧ (𝜃 → (𝑤𝑅𝑧 → ¬ 𝑤𝐶)))) ↔ ∀𝑤((𝜑𝜓) → (𝑧𝐶 ∧ ((𝜃 → (𝑤𝑅𝑧 → ¬ 𝑤𝐶)) ∧ (𝜃 → ((𝑤𝑅𝑧 → ¬ 𝑤𝐶) → (𝑤𝑅𝑧 → ¬ 𝑤𝐵)))))))
2322exbii 1844 . . . . 5 (∃𝑧𝑤((𝜑𝜓) → (𝑧𝐶 ∧ (𝜃 → (𝑤𝑅𝑧 → ¬ 𝑤𝐶)))) ↔ ∃𝑧𝑤((𝜑𝜓) → (𝑧𝐶 ∧ ((𝜃 → (𝑤𝑅𝑧 → ¬ 𝑤𝐶)) ∧ (𝜃 → ((𝑤𝑅𝑧 → ¬ 𝑤𝐶) → (𝑤𝑅𝑧 → ¬ 𝑤𝐵)))))))
241, 23mpbi 232 . . . 4 𝑧𝑤((𝜑𝜓) → (𝑧𝐶 ∧ ((𝜃 → (𝑤𝑅𝑧 → ¬ 𝑤𝐶)) ∧ (𝜃 → ((𝑤𝑅𝑧 → ¬ 𝑤𝐶) → (𝑤𝑅𝑧 → ¬ 𝑤𝐵))))))
25 imdi 393 . . . . . . . 8 ((𝜃 → ((𝑤𝑅𝑧 → ¬ 𝑤𝐶) → (𝑤𝑅𝑧 → ¬ 𝑤𝐵))) ↔ ((𝜃 → (𝑤𝑅𝑧 → ¬ 𝑤𝐶)) → (𝜃 → (𝑤𝑅𝑧 → ¬ 𝑤𝐵))))
26 pm3.35 801 . . . . . . . 8 (((𝜃 → (𝑤𝑅𝑧 → ¬ 𝑤𝐶)) ∧ ((𝜃 → (𝑤𝑅𝑧 → ¬ 𝑤𝐶)) → (𝜃 → (𝑤𝑅𝑧 → ¬ 𝑤𝐵)))) → (𝜃 → (𝑤𝑅𝑧 → ¬ 𝑤𝐵)))
2725, 26sylan2b 595 . . . . . . 7 (((𝜃 → (𝑤𝑅𝑧 → ¬ 𝑤𝐶)) ∧ (𝜃 → ((𝑤𝑅𝑧 → ¬ 𝑤𝐶) → (𝑤𝑅𝑧 → ¬ 𝑤𝐵)))) → (𝜃 → (𝑤𝑅𝑧 → ¬ 𝑤𝐵)))
2827anim2i 618 . . . . . 6 ((𝑧𝐶 ∧ ((𝜃 → (𝑤𝑅𝑧 → ¬ 𝑤𝐶)) ∧ (𝜃 → ((𝑤𝑅𝑧 → ¬ 𝑤𝐶) → (𝑤𝑅𝑧 → ¬ 𝑤𝐵))))) → (𝑧𝐶 ∧ (𝜃 → (𝑤𝑅𝑧 → ¬ 𝑤𝐵))))
2928imim2i 16 . . . . 5 (((𝜑𝜓) → (𝑧𝐶 ∧ ((𝜃 → (𝑤𝑅𝑧 → ¬ 𝑤𝐶)) ∧ (𝜃 → ((𝑤𝑅𝑧 → ¬ 𝑤𝐶) → (𝑤𝑅𝑧 → ¬ 𝑤𝐵)))))) → ((𝜑𝜓) → (𝑧𝐶 ∧ (𝜃 → (𝑤𝑅𝑧 → ¬ 𝑤𝐵)))))
3029alimi 1808 . . . 4 (∀𝑤((𝜑𝜓) → (𝑧𝐶 ∧ ((𝜃 → (𝑤𝑅𝑧 → ¬ 𝑤𝐶)) ∧ (𝜃 → ((𝑤𝑅𝑧 → ¬ 𝑤𝐶) → (𝑤𝑅𝑧 → ¬ 𝑤𝐵)))))) → ∀𝑤((𝜑𝜓) → (𝑧𝐶 ∧ (𝜃 → (𝑤𝑅𝑧 → ¬ 𝑤𝐵)))))
3124, 30bnj101 31993 . . 3 𝑧𝑤((𝜑𝜓) → (𝑧𝐶 ∧ (𝜃 → (𝑤𝑅𝑧 → ¬ 𝑤𝐵))))
32 ancl 547 . . . . 5 (((𝜑𝜓) → (𝑧𝐶 ∧ (𝜃 → (𝑤𝑅𝑧 → ¬ 𝑤𝐵)))) → ((𝜑𝜓) → ((𝜑𝜓) ∧ (𝑧𝐶 ∧ (𝜃 → (𝑤𝑅𝑧 → ¬ 𝑤𝐵))))))
33 bnj256 31976 . . . . 5 ((𝜑𝜓𝑧𝐶 ∧ (𝜃 → (𝑤𝑅𝑧 → ¬ 𝑤𝐵))) ↔ ((𝜑𝜓) ∧ (𝑧𝐶 ∧ (𝜃 → (𝑤𝑅𝑧 → ¬ 𝑤𝐵)))))
3432, 33syl6ibr 254 . . . 4 (((𝜑𝜓) → (𝑧𝐶 ∧ (𝜃 → (𝑤𝑅𝑧 → ¬ 𝑤𝐵)))) → ((𝜑𝜓) → (𝜑𝜓𝑧𝐶 ∧ (𝜃 → (𝑤𝑅𝑧 → ¬ 𝑤𝐵)))))
3534alimi 1808 . . 3 (∀𝑤((𝜑𝜓) → (𝑧𝐶 ∧ (𝜃 → (𝑤𝑅𝑧 → ¬ 𝑤𝐵)))) → ∀𝑤((𝜑𝜓) → (𝜑𝜓𝑧𝐶 ∧ (𝜃 → (𝑤𝑅𝑧 → ¬ 𝑤𝐵)))))
3631, 35bnj101 31993 . 2 𝑧𝑤((𝜑𝜓) → (𝜑𝜓𝑧𝐶 ∧ (𝜃 → (𝑤𝑅𝑧 → ¬ 𝑤𝐵))))
37 df-bnj17 31957 . . . . 5 ((𝜑𝜓𝑧𝐶 ∧ (𝜃 → (𝑤𝑅𝑧 → ¬ 𝑤𝐵))) ↔ ((𝜑𝜓𝑧𝐶) ∧ (𝜃 → (𝑤𝑅𝑧 → ¬ 𝑤𝐵))))
3837imbi2i 338 . . . 4 (((𝜑𝜓) → (𝜑𝜓𝑧𝐶 ∧ (𝜃 → (𝑤𝑅𝑧 → ¬ 𝑤𝐵)))) ↔ ((𝜑𝜓) → ((𝜑𝜓𝑧𝐶) ∧ (𝜃 → (𝑤𝑅𝑧 → ¬ 𝑤𝐵)))))
3938albii 1816 . . 3 (∀𝑤((𝜑𝜓) → (𝜑𝜓𝑧𝐶 ∧ (𝜃 → (𝑤𝑅𝑧 → ¬ 𝑤𝐵)))) ↔ ∀𝑤((𝜑𝜓) → ((𝜑𝜓𝑧𝐶) ∧ (𝜃 → (𝑤𝑅𝑧 → ¬ 𝑤𝐵)))))
4039exbii 1844 . 2 (∃𝑧𝑤((𝜑𝜓) → (𝜑𝜓𝑧𝐶 ∧ (𝜃 → (𝑤𝑅𝑧 → ¬ 𝑤𝐵)))) ↔ ∃𝑧𝑤((𝜑𝜓) → ((𝜑𝜓𝑧𝐶) ∧ (𝜃 → (𝑤𝑅𝑧 → ¬ 𝑤𝐵)))))
4136, 40mpbi 232 1 𝑧𝑤((𝜑𝜓) → ((𝜑𝜓𝑧𝐶) ∧ (𝜃 → (𝑤𝑅𝑧 → ¬ 𝑤𝐵))))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 398  wo 843  w3a 1083  wal 1531   = wceq 1533  wex 1776  wcel 2110  cin 3934   class class class wbr 5065  w-bnj17 31956   trClc-bnj18 31964
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1907  ax-6 1966  ax-7 2011  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2157  ax-12 2173  ax-ext 2793
This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1536  df-ex 1777  df-nf 1781  df-sb 2066  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-v 3496  df-in 3942  df-bnj17 31957
This theorem is referenced by:  bnj1190  32280
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